Download Statistical Mechanics to Disordered Quantum Optimization

Statistical Mechanics of Disordered
Quantum Optimization
Christopher Richard Laumann
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: S. L. Sondhi
September 2010
c Copyright by Christopher Richard Laumann, 2010.
All rights reserved.
Abstract
The classical statistical mechanical approach to complexity theory proceeds from
the study of ensembles of computationally intractable optimization problems as a
species of unusual disordered magnetic systems. Over the last thirty years, researchers
have used this approach to supplement worst-case hardness results encoded by complexity theory with detailed information about thermodynamic and dynamic phase
transitions in the structure of typical cases. This exchange of ideas between classical
statistical mechanics and computer science enabled the development of important
heuristic algorithms such as simulated annealing and survey propagation and further
refined our understanding of glassiness and critical slowing in physical disordered
systems.
In this thesis, we map out an analogous program in the quantum context. The
question is simple: what can quantum statistical mechanics reveal about the difficulty
of solving hard quantum optimization problems? Or more directly, what makes those
problems hard even for quantum computers? In this pursuit, we introduce the study
of ensembles of optimization problems whose complexity status is intrinsically quantum mechanical (Part I) and develop techniques to study quantum spin glasses and
the transverse field adiabatic algorithm applied to classically hard random optimization problems (Part II). In particular, we introduce the study of random quantum
satisfiability (QSAT) and identify the coarse aspects of its phase diagram, including
a new form of entanglement transition. We generalize the cavity method to the study
of quantum models and use it to study the transverse field Ising glass and frustrated
AKLT models on the Bethe lattice. We further apply the cavity method to extract
Griffiths-McCoy singularities in a diluted (classical) ferromagnet and finally observe
that there are no Goldstone bosons on the Bethe lattice.
iii
Acknowledgements
First, I would like to thank my advisor, Shivaji Sondhi, a condensed matter theorist, who, when I arrived at Princeton looking to work on topics related to quantum
computation, was not working on quantum computation with less vehemence than
anybody else. Since then, his curiosity, good humor and keen insight have guided us
unerringly in our exploration of unfamiliar topics in computation and glass theory.
At the same time, he managed to convey to me the beauty of the more traditional
questions of quantum many-body physics. I discovered that I was not working on
condensed matter with less vehemence than I expected and that I too might get
excited about dimers.
During my graduate years, I have had the immense good fortune to collaborate
with and learn from a peerless collection of physicists. I have spent many happy
hours at the blackboard arguing physics with Antonello Scardicchio, who introduced
me to cavity techniques and with whom I worked on my first project after coming
to Princeton and on many fruitful projects since. Likewise, Roderich Moessner has
become an indispensible ally in the battle against hard problems – and the battle
against Shivaji’s longer humorous digressions. I have learned much from Boris Spivak; most notably that it is somehow possible to know the answers before you start,
a skill that I have not yet mastered. I must thank Francesco Zamponi for his valiant
attempts to teach Shivaji and I about replica symmetry breaking and survey propagation and Andreas Läuchli for his amazing ability to turn questions into results
overnight. Finally, I thank my partner-in-crime, Siddharth Parameswaran, for many
intense physics conversations and the subsequent painful joint revision sessions. The
resulting work speaks for itself.
I would especially like to thank David Huse whose intellectual curiosity, insight
and patience know no bounds. Every working conversation and classroom lecture has
been a pleasure. I will finish those projects! I would also like to thank the other
iv
members of my committee, Curtis Callan and Jason Petta, who have both been kind
enough to take an interest in my work over the years. I must thank Jason especially for
taking me on as an experimental project student before he even arrived at Princeton.
I learned a great deal from that summer of fabrication and lab work about quantum
dots, semiconductors and real physics.
I cannot hope to possibly thank all of the scientists whose insights, discussions and
hospitality I have benefitted from over the last few years and I am sure that I have left
many out. Nonetheless, I would like to particularly thank M. Aizenmann, I. Arad, S.
Arora, A. Bernevig, R. Bhatt, S. Bravyi, E. Farhi, D. Haldane, S. Hughes, E. Lieb, A.
Ludwig, D. Nagaj, A. Obus, T. Osborne, G. Refael, O. Sattath, B. Terhal, S. Trebst,
U. Vazirani, F. Verstraete, and R. Zecchina. I would also like to especially thank
Laurel Lerner, whose indispensable support as friend and graduate administrator has
made my experience in Princeton immeasurably better.
For their unceasing support and love, I thank my family and in particular my
parents, Ed and Anne. Also, my brother Tim for always being up for long phone
conversations on whatever topics serve the hour, from neuroscience to art history.
Likewise, my other siblings, Eric and Lisa, and their families, for their motivational
queries about when I would finish.
I thank all of the wonderful friends I have made while here at Princeton. Coming
to central Jersey from city life was quite an adjustment for me, and it was only
through them that I found my place. In particular, I would like to thank my longtime housemates and friends Charles Mathy, Alex Barmas, Matthew Harrison, Ana
Ortiz and Fabio Rocha; my not-quite housemates Kristen Harkness, Katerina Visnjic,
Maria Martynovsky, Abhi Nellore, Fiona Burnell and Aakash Pushp; and, my former
daily co-conspirator, Eric Fountain. Last, and certainly not least, I thank Anushya
Chandran for the monkey, the owl and all the peaches.
v
To my parents.
vi
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
1 Overview of thesis
1
I
4
Random quantum satisfiability
2 Complexity theory for physicists
5
2.1
Problems, instances, computers and algorithms
. . . . . . . . . . . .
6
2.2
Polynomial reductions . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3
Classical: P and NP . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.4
Quantum: BQP and QMA . . . . . . . . . . . . . . . . . . . . . . . .
13
2.5
NP-Completeness: Cook-Levin . . . . . . . . . . . . . . . . . . . . . .
16
2.5.1
3-SAT is in NP . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.5.2
3-SAT is NP-complete . . . . . . . . . . . . . . . . . . . . . .
17
QMA-Completeness: Kitaev . . . . . . . . . . . . . . . . . . . . . . .
20
2.6.1
QSAT is QMA1 . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.6.2
QSAT is QMA1 -complete . . . . . . . . . . . . . . . . . . . . .
22
Typical versus worst-case complexity . . . . . . . . . . . . . . . . . .
27
2.6
2.7
vii
2.8
Classical statistical mechanics of k-SAT . . . . . . . . . . . . . . . . .
28
2.9
Phase diagram of classical random k-SAT
. . . . . . . . . . . . . . .
31
2.10 Adiabatic quantum algorithm for classical k-SAT . . . . . . . . . . .
33
3 Random Quantum Satisfiability
3.1
36
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.1.1
Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2-QSAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2.1
Trees are SAT . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.2
Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.2.3
Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.2.4
Extensivity and the promise gap . . . . . . . . . . . . . . . . .
45
Geometric Properties of QSAT . . . . . . . . . . . . . . . . . . . . . .
47
3.3.1
Geometrization theorem . . . . . . . . . . . . . . . . . . . . .
48
3.3.2
Product satisfiability . . . . . . . . . . . . . . . . . . . . . . .
49
3.3.3
Counting product states at M = N . . . . . . . . . . . . . . .
54
Random k-QSAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.4.1
Existence of PRODSAT . . . . . . . . . . . . . . . . . . . . .
57
3.4.2
Existence of UNSAT . . . . . . . . . . . . . . . . . . . . . . .
60
3.4.3
Existence of SAT-UNPRODSAT . . . . . . . . . . . . . . . . .
62
3.4.4
Satisfying the promise . . . . . . . . . . . . . . . . . . . . . .
68
3.4.5
QSAT at rank r > 2 . . . . . . . . . . . . . . . . . . . . . . .
69
Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.A Formal PRODSAT proof . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.B Reduced density matrix diagnostic for product states . . . . . . . . .
72
3.2
3.3
3.4
3.5
viii
II
Cavity method and the Bethe lattice
75
4 Introduction to the cavity method
80
4.1
The Bethe-Peierls method for the Ising ferromagnet . . . . . . . . . .
82
4.2
Classical disordered magnet on a Cayley tree . . . . . . . . . . . . . .
85
4.3
Cayley trees versus regular random graphs . . . . . . . . . . . . . . .
88
4.4
Relation to belief propagation . . . . . . . . . . . . . . . . . . . . . .
90
4.5
Quantum cavity method . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.6
Population dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
5 Transverse field Ising glass
98
5.1
Model and cavity solution . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1
5.3
5.4
99
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.1
Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.2
Structure of the glassy phase . . . . . . . . . . . . . . . . . . . 109
Discussion and further work . . . . . . . . . . . . . . . . . . . . . . . 112
6 AKLT models with quantum spin glass ground states
117
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2
AKLT states: a brief review . . . . . . . . . . . . . . . . . . . . . . . 120
6.3
Transfer matrix solution of the classical problem on trees . . . . . . . 122
6.4
Variational bounds on the gap . . . . . . . . . . . . . . . . . . . . . . 127
6.5
AKLT model on regular random graphs: frustration and the spin glass
state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.6
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.A Transfer matrix for the AKLT model . . . . . . . . . . . . . . . . . . 139
6.B Stability against spin glass ordering on a regular random graph
ix
. . . 140
7 Griffiths-McCoy singularities and Lee-Yang zeros in a solvable diluted ferromagnet
142
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2
Model and organization . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.3
Phase diagram and cavity equations . . . . . . . . . . . . . . . . . . . 145
7.4
Cluster series at J = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.5
Cavity approach at J = ∞ . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6
Integral representation for the magnetization . . . . . . . . . . . . . . 157
7.7
Density of Lee-Yang zeros at J = ∞ . . . . . . . . . . . . . . . . . . 160
7.8
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.9
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8 There are no Goldstone bosons on the Bethe lattice
168
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.2
The classical spherical model . . . . . . . . . . . . . . . . . . . . . . . 171
8.3
8.4
8.2.1
Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.2.2
Ordered phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.2.3
Spatial correlations . . . . . . . . . . . . . . . . . . . . . . . . 175
The quantum spherical model . . . . . . . . . . . . . . . . . . . . . . 176
8.3.1
Quantum phase transition . . . . . . . . . . . . . . . . . . . . 178
8.3.2
Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Bibliography
184
x
List of Tables
3.1
Critical values for random k-(Q)SAT. . . . . . . . . . . . . . . . . . .
xi
60
List of Figures
2.1
NP verification circuit. . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
Interpretation of Boolean AND gate as three-body interaction. . . . .
19
2.3
QMA verification circuit. . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
Examples of interaction graphs for (a) 2-SAT and (b) 3-SAT. . . . . .
30
2.5
Clustering transitions in random CSPs. . . . . . . . . . . . . . . . . .
31
3.1
Phase diagram of k-QSAT. . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2
Phase diagram of 2-QSAT. . . . . . . . . . . . . . . . . . . . . . . . .
44
3.3
Dimer coverings of 3-QSAT. . . . . . . . . . . . . . . . . . . . . . . .
51
3.4
Rank analysis of reduced density matrices in a SAT state. . . . . . .
66
3.5
Finite size diagonalization of random k-QSAT. . . . . . . . . . . . . .
68
3.6
Finite size scaling of αc for k-QSAT. . . . . . . . . . . . . . . . . . .
69
4.1
Iteration, merging and link addition. . . . . . . . . . . . . . . . . . .
83
4.2
q = 3 regular random graph with a cavity. . . . . . . . . . . . . . . .
91
4.3
Belief propagation on a regular random graph. . . . . . . . . . . . . .
92
5.1
Iteration of cavity rods. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2
Merging of cavity rods onto “broken” rod. . . . . . . . . . . . . . . . 105
5.3
Phase diagram of ±J transverse field Ising glass. . . . . . . . . . . . . 110
5.4
Statistics at the glass transition. . . . . . . . . . . . . . . . . . . . . . 110
5.5
Finite size phase transition curves. . . . . . . . . . . . . . . . . . . . 111
xii
5.6
Distribution of cavity fields. . . . . . . . . . . . . . . . . . . . . . . . 113
5.7
Distribution of cavity interactions. . . . . . . . . . . . . . . . . . . . . 114
5.8
Long range cavity interactions.
6.1
Phase diagram of Bethe lattice AKLT model. . . . . . . . . . . . . . 132
7.1
Phase diagram of diluted ferromagnet on q = 3 Bethe lattice. . . . . . 145
7.2
Lyapunov exponents of ferromagnetic fixed point at finite J. . . . . . 149
7.3
Analytic structure of integrand of equation (7.35). . . . . . . . . . . . 159
7.4
Harmonic expansion of density of Lee-Yang zeros. . . . . . . . . . . . 162
7.5
Density of Lee-Yang zeros at and above criticality at J = ∞. . . . . . 162
7.6
Cavity field distribution of DFM at criticality. . . . . . . . . . . . . . 165
7.7
Cavity field distribution at finite J critical point. . . . . . . . . . . . 165
7.8
Critical magnetization at J = ∞. . . . . . . . . . . . . . . . . . . . . 166
8.1
Support of spectral response of Gij . . . . . . . . . . . . . . . . . . . . 180
8.2
Analytic structure of Gij (ω). . . . . . . . . . . . . . . . . . . . . . . . 181
. . . . . . . . . . . . . . . . . . . . . 115
xiii
Chapter 1
Overview of thesis
The classical statistical mechanical approach to complexity theory proceeds from the
study of ensembles of computationally intractable optimization problems as a species
of unusual disordered magnetic systems. Over the last thirty years, researchers have
used this approach to supplement worst-case hardness results encoded by complexity
theory with detailed information about thermodynamic and dynamic phase transitions in the structure of typical cases. This exchange of ideas between classical statistical mechanics and computer science enabled the development of important heuristic
algorithms such as simulated annealing and survey propagation and further refined
our understanding of glassiness and critical slowing in physical disordered systems.
In this thesis, we map out an analogous program in the quantum context. The
question is simple: what can quantum statistical mechanics reveal about the difficulty
of solving hard quantum optimization problems? Or more directly, what makes those
problems hard even for quantum computers? In this pursuit, we introduce the study of
ensembles of optimization problems whose complexity status is intrinsically quantum
mechanical (Part I) and develop techniques to study quantum spin glasses and the
transverse field adiabatic algorithm applied to classically hard random optimization
problems (Part II).
1
Thus, in Chapter 2, we review the classical complexity theory necessary to understand the important statement that P 6= NP and its more recent quantum generalization BQP 6= QMA. These complexity theoretic conjectures essentially assert
that there exist natural classes of problems (called NP-complete and QMA-complete)
which cannot in general be solved efficiently by any computational process. This does
not rule out the possibility that particular, or even typical, instances of such problems
are straightforward. This leads to the study of typical-case complexity for random
ensembles of NP-complete problems – ensembles which bear many of the features
of mean field spin glasses. We will review some of the qualitative features of this
connection in the context of classical optimization.
In Chapter 3, we introduce the study of random instances of the quantum generalization of satisfiability (QSAT). This problem is QMA1 -complete and expected
to be hard for quantum computers to solve in general. Thus, one might hope that
many of the classical glass phenomena that arise in the study of random classical
hard problems have quantum analogs. Indeed, work over the past several years has
pushed the understanding of the QSAT phase diagram quite far and has established
an entanglement transition – up to some critical clause density the typical instance of
QSAT has satisfying product states while beyond this density, QSAT is still satisfiable
but only by entangled, non-product, states. If further progress on the structure of the
ground state space can be made, it may lend itself to the development of quantum
counterparts to the heuristic classical algorithms.
Many of the sharpest results regarding classical optimization problems follow from
cavity analyses and thus we would very much like to develop such methods in the
quantum context. This is the focus of Part II. The cavity method refers broadly to
a cluster of analytic techniques for treating spin models on locally tree-like graphs
through self-consistent local recursion relations. Such approaches date back to the
early part of the twentieth century when Bethe and Peierls introduced their mean field
2
theory of ferromagnetism. Suitably dressed to handle disordered models, modern cavity techniques, á la Mezard and Parisi, have proven extremely useful for understanding both classical optimization and the phenomena associated with replica symmetry
breaking in mean field spin glasses. Chapter 4 provides a more detailed introduction
of the relevant aspects of cavity analysis and points out a general framework for lifting
these techniques to quantum mechanical models.
In Chapter 5, we develop a quantum cavity analysis of the transverse field Ising
glass on the Bethe lattice which allows us to ascertain its phase diagram and many of
the properties of the deep quantum glassy phase. The combined analytic-numerical
approach that we propose is quite general and, since its publication, has motivated
work by other groups on a number of related models on tree-like graphs – the transverse field ferromagnet, the Bose Hubbard model and, most recently, the Bose glass.
We have ourselves studied both the frustrated AKLT (Chapter 6) and bosonic spherical models (Chapter 8) on such graphs in order to illuminate some of the strange
phenomena associated with quantum phase transitions in such mean-field-esque long
range models. Finally, we include a treatment of the classical diluted ferromagnet
by the cavity method in Chapter 7 as an illustration of the power of the local cavity
analysis to extract even delicate features of statistical models such as Griffiths-McCoy
singularities and the density of complex Lee-Yang zeros.
3
Part I
Random quantum satisfiability
4
Chapter 2
Complexity theory for physicists
Complexity theory classifies how “hard” it is to compute the solution to a problem
as a function of the input size N of the problem instance. In particular, algorithms
are considered efficient if the amount of time they take to complete scales at most
polynomially with the size of the input and inefficient otherwise. The classification of
algorithms by asymptotic efficiency up to polynomial transformations is the key to the
robustness of complexity theoretic results. Using these definitions, the classification of
problem hardness is essentially independent of the underlying model of computation
and suggests a physical significance to the abstract concept of hardness.
In this line of reasoning, if P 6= NP, as is the current consensus, there are natural
classes of problems which cannot be solved in polynomial time by any computational
process, including any physical process which can be simulated by computer. From a
physical point of view, this amounts to a ‘no-go’ theorem for certain kinds of dynamics
that arise in the simulation of these hard problems. As physicists, we might ask what
makes such problems hard or what kinds of dynamics must be slow. The problem
with this approach is that the class of all possible instances of hard problems is
too broad to get a good handle on. Luckily, complexity theory provides its own
guide to focusing our attention: it turns out that certain NP problems, those termed
5
NP-complete, capture the full hardness of the class NP. In particular, the problem
of boolean satisfiability of 3-bit clauses, 3-SAT, is NP-complete and therefore can
encode the full hardness of the class NP.
The advent of the quantum computer modifies the above reasoning only slightly.
It appears that quantum computers are somewhat more powerful than their classical counterparts, so that we must introduce new quantum complexity classes to
characterize them. Nonetheless, analogous statements hold within this new framework: quantum polynomial (BQP) is larger than classical polynomial (P) but not
powerful enough to contain classical verifiable (NP), nor quantum verifiable (QMA).
If we wish to study the hardest of quantum problems, we may turn to the study of
QMA-complete problems such as LOCAL HAMILTONIAN and the closely related
QSAT.
In the first half of this chapter, we will provide a concise review of the key concepts for the above argument culminating in a discussion of worst-case hardness and
the Cook-Levin theorem, showing the existence of NP-complete problems, and the
quantum analogues due to Kitaev and Bravyi. This story motivates and complements
the statistical study of ‘typical’ instances of 3-SAT, 3-QSAT and other classical and
quantum hard optimization problems, which will be discussed in the later sections
and following chapters.
2.1
Problems, instances, computers and algorithms
The success of complexity theory as a classification scheme for the “hardness” of
problems is in part due to the careful definitions employed. Here we will sketch the
most important aspects of these concepts and leave the rigorous formalism for the
textbooks, of which we particularly recommend Arora and Barak to the interested
6
reader [11]. We have taken a particular path through the forest of variations on the
ideas below and do not pretend to completeness.
Throughout this introduction we will focus on so-called decision problems, that
is Yes/No questions such as “Does the Hamiltonian H have a state with energy less
than E?” rather than more general questions like “What is the ground state energy
of H?” This restriction is less dramatic than it seems – many general questions may
be answered by answering a (reasonably short) sequence of related Yes/No questions,
much like playing the game twenty questions – and it significantly simplifies the conceptual framework we need to introduce. Moreover, many of the essential complexity
theoretic results arise already within the context of decision problems.
A decision problem, then, is a question that one may ask of a class of instances.
For example, the DIVIDES problem asks “Does a divide b?” for integers a and b.
Leaving the variables a and b unspecified, we clearly cannot yet answer this question.
An instance of DIVIDES might be “Does 5 divide 15?” A moment’s thought now
reveals that a definitive answer exists: Yes. We refer to instances of a problem as
Yes-instances (No-instances) if the answer is Yes (No). We follow computer science
convention by giving problems fully capitalized names.1
What does it mean to solve a problem? We would not feel we had solved the
problem if we could only answer a few specific instances. On the other hand, we
certainly could not expect to have a book containing the (infinite) table of answers
to all possible instances for easy reference. Thus, we want a general algorithm which,
given an arbitrary instance, provides us with a step-by-step recipe by which we can
decide the instance. Some physical object must carry out the algorithmic instructions
and it is this object that we call a computer – whether it is a laptop running C code,
ions resonating in an ion trap or a sibling doing long division with pencil and paper.
Thus, a solution to a decision problem is an algorithm which can decide arbitrary
1
We trust this will not give physics readers PROBLEMS.
7
instances of the problem when run on an appropriate computer. Such an algorithm
for a decision problem is often called a decision procedure.
Clearly it is less work to answer the DIVIDES problem for small numbers than
for large. “Does 5 divide 15?” takes essentially no thought at all while “Does 1437
divide 53261346150?” would take a few moments to check. We therefore define the
input size (or just size) of an instance as the number of symbols we need to specify
the instance. In the DIVIDES problem, we could take the size as the number of
symbols needed to specify the pair (a, b). The size N of (5, 15) would be 6 while that
of (1437, 53261346150) is 18.
Computer scientists measure the efficiency of an algorithm by considering the
asymptotic scaling of its resource consumption with the input size of the problem.
More precisely, consider the finite but large collection of all possible problem instances
whose size is no greater than N . For each of these instances, the algorithm will take
some particular amount of time. For the finite collection at size N , there will be
a worst-case instance which takes more time T than any of the others at that size.
Complexity theory generally focusses on the scaling of this worst-case time T as a
function of input size N as N → ∞. Clearly, the slower the growth of T with N ,
the more efficient the algorithm for large inputs. Indeed, algorithmic procedures are
considered efficient so long as T grows at most polynomially with N , for any particular
polynomial we like. Thus both linear and quadratic growth are efficient, even though
linear is clearly faster. Anything slower, such as T = O(eN ), is inefficient.
For example, the most famous decision procedure for DIVIDES – long division –
takes of order T = O(log b × log a) ≤ O(N 2 ) arithmetic steps to perform the division
and check the remainder is 0. That T grows with a and b logarithmically corresponds
nicely to our intuition that bigger numbers are harder to divide, but not too much
harder. It is instructive to consider a different, inefficient, algorithm for the same
problem. Suppose we had not yet learned how to divide but knew how to multiply.
8
We might try the following decision procedure: try to multiply a with every number
c from 1 up to b and check if ac = b. This trial-and-error approach would take
T = O(log a × log b × b) ≈ O(ecN ) to try out all the possibilities up to b. Even
for relatively small instances, this approach would quickly become prohibitively time
consuming – simply enumerating all of the numbers of up to 30 digits at one per
nanosecond would take longer than the age of the universe!
Finally, we lift our classification of algorithm efficiency to a classification of problem hardness: A problem is tractable if there exists an efficient algorithm for solving
it and it is intractable otherwise. By this definition, DIVIDES is tractable (long division solves it efficiently), despite the existence of alternative slower algorithms. As
we will discuss further in the following sections, we can rarely prove that no efficient
algorithm exists for a given problem, but complexity theory nonetheless offers strong
arguments that certain large classes of problems are intractable in this sense.
Computers are clearly central to the determination of the difficulty of problems
– we classify problems according to the efficiency of the computational algorithms
that exist for treating them. In addition to the time taken, we can measure the resource requirements in various implementation dependent ways – memory consumed,
number of gates required, laser pulse bandwidth, quantity of liquid Helium evaporated. One might expect that the kind of computer that we use would greatly impact
any complexity classification. At the very least, your laptop will be faster than your
brother at dividing 1000 digit numbers. The beauty of the definition of efficiency by
polynomial scaling is that many of these implementation dependent details drop out
and we really can focus on the time efficiency as an overall measure of difficulty.2
2
In practice the amount of memory or the number of cores in a workstation regularly limits
its ability to do computations. Since in finite time, even a parallel computer can only do a finite
amount of work or address a finite amount of memory, a polynomial bound on T also provides a
polynomial bound on the space requirements. Likewise, finite parallelization only provides constant
time improvements. More refined classifications can be made by restricting resource consumption
more tightly but we will not consider them here.
9
The robustness of these definitions follows from one of the great ideas of complexity theory: up to polynomial overheads, any reasonable classical computer may be
simulated by any other. This is known as the strong Church-Turing hypothesis and,
as its name suggests, is only a conjecture. Nonetheless, it has been examined and
confirmed for many particular models of classical computation3 and is widely believed
to hold more generally. This is the reason for defining efficiency up to polynomial
scaling: since any computer can simulate the operation of any other up to polynomial
overheads, all computers can solve the same problems efficiently. In Sec. 2.3 below,
we will consider the most important classical complexity classes that arise from these
coarse but robust definitions of efficiency.
The careful reader will have noticed that we restricted our statement of the
Church-Turing hypothesis to classical computers. It is widely believed that classical
computers cannot efficiently simulate quantum systems. Certainly, directly simulating Schrödinger’s equation on a polynomially sized classical computer is problematic
since the Hilbert space is exponentially large. On the other hand, if we had a quantum
computer with which to do our simulation, the state space of our computer would also
be a Hilbert space and we could imagine representing and evolving complex states
of the system by complex states and evolutions of the quantum computer. This reasoning leads to the strong quantum Church-Turing hypothesis: that any reasonable
quantum computer may be efficiently simulated by any other. With this hypothesis
in hand, we may proceed to develop a robust classification of quantum complexity
classes, as in Sec. 2.4.
2.2
Polynomial reductions
Reduction is the most important tool in complexity theory. A decision problem A
reduces to another problem B if there is a polynomial time algorithm which can
3
For example, Turing machines, Boolean circuit models and your sibling.
10
transform instances of A into instances of B such that Yes-instances (No-instances)
of A map to Yes-instances (No-instances) of B. In this case, B is at least as hard as
A: any algorithm which could efficiently decide B would be able to efficiently decide
A as well. Just use the transformation to convert the given instance of A into an
instance of B and then apply the efficient algorithm for B.
Reductions formalize the interrelationships between problems and allow us to show
that new problems are actually part of known classes. Obviously, if we can reduce a
problem A to a problem B that we know how to solve efficiently, we have just shown
how to solve A efficiently as well. Conversely, if we have a problem C which we believe
is intractable – that is, not solvable by an efficient algorithm – and we can reduce it
to another problem D, that suggests D should also be intractable. Using this logic
we can try to show that all kinds of interesting problems ought to be intractable if
we can find one to start with.
2.3
Classical: P and NP
The most important complexity class is known as P – this is the class of decision
problems which a classical computer can decide efficiently. More precisely, a decision
problem is in P if there exists an algorithm that runs in polynomial time as a function
of the input size of the instance and outputs Yes or No depending on whether the
instance is a Yes-instance or No-instance of the problem. From a logical point of
view, to show that a given problem is in P we need to provide an efficient procedure
to decide arbitrary instances. Colloquially, P is the class of problems that are easy
to solve.
We have already discussed one example, the DIVIDES problem, for which long
division constitutes a polynomial time decision procedure. Another example is given
by the energy evaluation problem: ”Does a specific configuration σ of a classical Ising
11
Hamiltonian H =
P
Jij σi σj have energy less than E?” Here the instance is specified
by a configuration made of N bits, a Hamiltonian function with up to N 2 coupling
terms and a threshold energy E (where all real numbers are specified with some fixed
precision). Since we can evaluate the energy H(σ) using of order N 2 multiplications
and additions and then compare it to E, this problem is also in P.
The second most important complexity class is NP: this is the class of decision
problems for which there exists a scheme by which Yes-instances may be efficiently
verified by a classical computation. We may think of this definition as a game between
a prover and a verifier in which the prover attempts, by hook or by crook, to convince
the verifier that a given instance is a Yes-instance. The prover provides the verifier
with a proof of this claim which the verifier can efficiently check and either Accept or
Reject. NP places no restrictions on the power of the prover – only that Yes-instances
must have Acceptable proofs, No-instances must not have Acceptable proofs and that
the verifier can decide the Acceptability of the proof efficiently. We note that there
is an intrinsic asymmetry in the definition of NP: we do not need to be able to verify
that a No-instance is a No-instance.
For example, the problem “Does the Hamiltonian H =
P
Jij σi σj have a ground
state with energy less than E?” has such an efficient verification scheme. If the
prover wishes to show that a given H has such low energy states, he can prove it to
the verifier by providing some configuration σ which he claims has energy less than
E. The skeptical verifier may efficiently evaluate H(σ) using the algorithm outlined
above and if indeed H(σ) < E, the skeptic would Accept the proof. If H did not have
such low energy states, then no matter what the prover tried, he would be unable to
convince the verifier to Accept.
At first brush, NP seems a rather odd class – why should we be so interested in
problems whose Yes-instances may be efficiently checked? Of course, any problem we
can decide efficiently (in P) can be checked efficiently (because we can simply decide
12
it!). What of problems outside of NP? These do not admit efficient verification
schemes and thus certainly cannot have efficient decision procedures. Moreover, even
if, by some supernatural act of intuition (not unusual in theoretical physics), we guess
the correct answer to such a problem, we would not be able to convince anybody else
that we were correct. There would be no efficiently verifiable proof! Thus, NP is the
class of problems that we could ever hope to be convinced about.
Since 1971, the outstanding question in complexity theory (worth a million dollars
since the new millennium), has been “Is P = NP?” This would be an astonishing
result: it would state that all of the difficulty and creativity required to come up with
the solution to a tough problem could be automated by a general purpose algorithm
running on a computer. The determination of the truth of theorems would be a
simple matter of asking your laptop to think about it. Since most scientists believe
that there are hard problems, beyond the capability of general purpose algorithms,
the consensus holds that P 6= NP.4
2.4
Quantum: BQP and QMA
The most important quantum complexity class is BQP – this is the class of decision
problems which a quantum computer can decide efficiently with bounded error (the
B in the acronym). Since general quantum algorithms have intrinsically stochastic
measurement outcomes, we have no choice but to allow for some rate of false-positive
and false-negative outcomes. As long as these rates are bounded appropriately (say
by 1/3), a few repetitions of the quantum computation will exponentially amplify the
probability of a determining the correct result. Thus, BQP is the quantum analogue of
4
This may of course be the bias of the scientists who don’t want to be replaced by omnipotent
laptops.
13
P and plays a similar role in the classification of decision problems. Since a quantum
computer can simulate any classical computation, P is contained in BQP.5
The most important example of a BQP problem that is not known to be in P
is integer factoring. As a decision problem, this asks “Given N and M , does the
integer N have a factor p with 1 < p ≤ M ?” In the 90’s, Peter Shor famously
proved that factoring is in BQP by developing a quantum factoring algorithm. There
is no proof that factoring is classically hard (outside of P) – nonetheless, many of
the cryptography schemes on which society relies for secure communication over the
internet are only secure if it is. Shor’s algorithm renders all of these schemes useless
if a large scale quantum computer is ever built.
The quantum analogue of NP is the class QMA, Quantum Merlin-Arthur, which
is the class of decision problems whose Yes-instances can be efficiently checked by a
quantum computer given a quantum state as a proof (or witness). The colorful name
comes from the description of this class in terms of a game: Merlin, all-powerful
but less than trustworthy, wishes to prove to Arthur, a fallible but well-intentioned
individual with a quantum computer, that a particular instance of a problem is a
Yes-instance. Merlin, using whatever supernatural powers he enjoys, provides Arthur
with a quantum state designed to convince Arthur of this claim. Arthur then uses
his quantum computer to decide, with some bounded error rate (say 1/3), whether
to accept or reject the proof.
There are three primary differences between NP and QMA: 1) the verifier is a
quantum computer, 2) the proof is a quantum state, and, 3) the verification is allowed
a bounded error rate. The first two differences provide the class with its additional
quantum power; that the verifier is allowed a bounded error rate is necessary due
to quantum stochasticity, but not believed to be the source of its additional power.
5
For the expert, we note that a closer analogue of BQP is BPP, the class of decision problems
which can be efficiently decided by a randomized classical algorithm with bounded error. In an
attempt to minimize the onslaught of three letter acronyms, we have left this complication out.
14
We note that the particular error bound is again somewhat arbitrary – Arthur can
exponentially improve the accuracy of a noisy verification circuit by requesting Merlin
provide him multiple copies of the proof state and running his verifier multiple times
[4]. Thus, even a verifier which falsely accepts No-instances with probability up to
p − 1/poly(N ) while accepting valid proofs with probability p only slightly larger can
be turned into an efficient bounded error QMA verifier through repetition.
An example of a QMA problem is given by the k-LOCAL HAMILTONIAN problem:
Input: A quantum Hamiltonian H =
P
m
Am composed of M bounded operators,
each acting on k qubits of an N qubit Hilbert space. Also, two energies a < b,
separated by at worst a polynomially small gap b − a > 1/poly(N ).
Output: Does H have an energy level below a?
Promise: Either H has an energy level below a or all states have energies above b.
Here we have introduced the notion of a ‘promise’ in a decision problem. Promises
are a new feature in our discussion: they impose a restriction on the instances that
a questioner is allowed to present to a decision procedure. The restriction arises
because the algorithms and verification procedures we use to treat promise problems
need not be correct when presented with instances that do not satisfy the promise
– an efficient solver for LOCAL HAMILTONIAN could in fact fail on Hamiltonians
with ground state energies in the promise gap between a and b and we would still
consider LOCAL HAMILTONIAN solved.
Heuristically, it is clear why we need the promise gap for LOCAL HAMILTONIAN
to be QMA: suppose we had a quantum verifier which took a quantum state |ψi and
tried to measure its energy = hψ|H|ψi through a procedure taking time T . Timeenergy uncertainty suggests that we should not be able to resolve to better than
15
1/T . Thus, if T is to be at most polynomially large in N , the verifier would not be
able to determine whether exponentially close to a is above or below the threshold.
The actual construction of a verification circuit for the LOCAL HAMILTONIAN
problem is somewhat more subtle than simply ‘measuring’ the energy of a given state.
As we will provide a very closely related construction for the QSAT problem below,
we do not include the verifier for LOCAL HAMILTONIAN. We refer the interested
reader to Ref. [4].
The quantum analogue of the classical claim that P 6= NP is that BQP 6= QMA –
a conjecture that is strongly believed for many of the same reasons as in the classical
case.
2.5
NP-Completeness: Cook-Levin
In the early 1970s, Cook and Levin independently realized that there are NP problems
whose solution captures the difficulty of the entire class NP. These are the so-called
NP-complete problems. What does this mean?
A problem is NP-complete if it is both a) in NP (efficiently verifiable) and b) any
problem in NP can be reduced to it efficiently. Thus, if we had an algorithm to solve
an NP-complete problem efficiently, we could solve any problem whatsoever in NP
efficiently. This would prove P = NP with all of the unexpected consequences this
entails. Assuming on the contrary that P 6= NP, any problem which is NP-complete
must be intractable.
Let us sketch a proof of the Cook-Levin theorem showing the existence of NPcomplete problems. In particular, we will show that classical 3-satisfiability, 3-SAT,
is NP-complete. 3-SAT is the decision problem which asks whether a given Boolean
expression composed of the conjunction of clauses, each involving at most 3 binary
variables, has a satisfying assignment. Re-expressed as an optimization problem,
16
3-SAT asks, “Does the energy function
H=
X
Em (σm1 , σm2 , σm3 ),
(2.1)
m
acting on N binary variables σi in which each local energy term Em takes values 0 or
1, have a zero energy (satisfying) ground state?”6
2.5.1
3-SAT is in NP
First, it is clear that 3-SAT is itself efficiently verifiable and therefore in NP. If a
prover wishes to prove that a particular instance H is satisfiable, she could provide
a verifier a zero energy configuration. The verifier would take this configuration and
evaluate its energy (using arithmetic in a linear number of steps) and thus be able
to check the validity of the claim. If H is satisfiable, such a configuration exists. On
the other hand, if H is not satisfiable, the prover would not be able to convince the
verifier otherwise because all states would have energy greater than zero.
2.5.2
3-SAT is NP-complete
The tricky part is to show that 3-SAT is as hard as the entire class NP. We need
to show that any possible NP problem can be reduced to a 3-SAT problem by a
polynomial transformation. What is the only thing that all NP problems have in
common? By definition, they all have polynomial size verification procedures which
take as input a proposed proof that an instance is a Yes-instance and output either
Accept or Reject based on whether the proof is valid. This verification procedure is
what we will use to provide the reduction to 3-SAT.
6
The interactions Em in 3-SAT are usually defined to penalize exactly one of the 23 = 8 possible
configurations of its input variables – but allow each of the N3 possible 3-body interactions to
appear multiple times in the sum. Thus, our definition is equivalent up to absorbing these terms
together, which modifies the excited state spectrum but not the counting of zero energy satisfying
states.
17
σ0,0
Proof
σ0,1
σ0,2
σT,0
Accept
σ0,N
t
0
1
2
T
Figure 2.1: Circuit representing an NP verifier. The circuit depends on the particular
instance and must be efficiently constructible by a polynomial time circuit drawing
algorithm.
Let us think of the verification procedure for a particular instance A of some NP
problem as a polynomially sized Boolean circuit as in Fig. 2.1. The input wires encode
the proposed proof that A is a Yes-instance and the output wire tells us whether to
Accept or Reject the proof. The gates in the figure are simply the usual Boolean logic
gates such as NAND and NOR, which take two input bits and provide one output
bit. Any Boolean circuit may be written using such binary operations with arbitrary
fan-out, so we assume that we can massage the verification circuit into the form
shown. Now we will construct an instance of 3-SAT encoding the operation of this
circuit. That is, if the instance is satisfiable, then there exists a proof that the verifier
accepts showing that the original NP problem is a Yes-instance and conversely, if the
instance is not satisfiable, then no such proof exists and the original NP problem is a
No-instance.
The 3-SAT instance is very simple to construct if we simply change our point of
view on the picture in Fig. 2.1. Instead of viewing it as a Boolean circuit operating
from left to right, let us view it as the interaction graph for a collection of O(N × T )
18
σ0 σ1 σ2 E
σ0
σ1
0
0
0
0
1
1
1
1
σ2
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
0
Figure 2.2: Interpretation of Boolean AND gate as three-body interaction.
binary bond variables – one for each of the wires in the circuit: the input bits of the
proof, the output bit and each of the intermediate variables. Each gate then specifies
a 3-body interaction Em for the adjacent variables which we define to take the value 0
for configurations in which the variables are consistent with the operation of the gate
and 1 otherwise. See Fig. 2.2. We now add a final 1-body term on the output bit of
the verification circuit which penalizes the Reject output. We now have an Ising-like
model with polynomially many 3-body interactions and non-negative energy.
That’s it. If the 3-SAT instance described by Fig. 2.1 has a zero energy ground
state, then there is a configuration of the wire variables such that the circuit operates
correctly and produces an Accept output. In this state, the input wires represent a
valid proof showing that the original instance was a Yes-instance. On the other hand,
if the 3-SAT instance is not satisfiable, no state exists such that the circuit operates
correctly and the output is Accept. Thus we have shown that all problems in NP can
be efficiently reduced to 3-SAT.
Now that we have one problem, 3-SAT, which is NP-complete, it is straightforward
to show the existence of many other NP-complete problems: we need only find reductions from 3-SAT to those problems. Indeed, a veritable menagerie of NP-complete
problems exists (see [52]) including such famous examples as the traveling salesman
problem and graph coloring. A more physics oriented example is that of determining
the ground state energy of the ±J Ising model in 3 or more dimensions [13].
19
2.6
QMA-Completeness: Kitaev
The complexity class QMA provides the quantum analogue to NP and, just like NP,
it contains complete problems which capture the difficulty of the entire class. Kitaev
first introduced the QMA-complete problem 5-LOCAL HAMILTONIAN in the early
’00s and proved its completeness using a beautiful idea due to Feynman: that of the
sum over histories. The quantum Cook-Levin proofs are somewhat more complicated
than the classical case and we will only sketch them here. For simplicity and to
connect with the statistical study undertaken in the next chapter, we will restrict
our attention to the slightly simpler problem of k-QSAT, which is QMA1 -complete
for k ≥ 4. QMA1 is the variant of QMA in which the verification error is one-sided:
Yes-instances may be verified with no errors while invalid proofs still occasionally get
incorrectly accepted.
We sketch the proof that QSAT is QMA1 -complete. The details may be found in
[22].
First, let us define k-QSAT a bit more carefully:
Input: A quantum Hamiltonian H =
P
m
Πm composed of M projectors, each acting
on at most k qubits of an N qubit Hilbert space.
Output: Does H have a zero energy ground state?
Promise: Either H has a zero energy state or all states have energy above a promise
gap energy ∆ > 1/poly(N ).
We note that QSAT is clearly as hard as classical SAT: classical SAT instances
are simply instances of QSAT whose projectors point at computational basis states.
The promise is trivially satisfied since classical SAT energies are integers and thus
either 0 or ≥ 1. In general, we expect QSAT to be even harder, since we believe that
it can only be verified with a quantum computer and thus not part of NP.
20
2.6.1
QSAT is QMA1
To show that QSAT is QMA1 , we need to find an efficient quantum verification scheme
such that (a) there exist proofs for Yes-instances which our verifier always accepts
and (b) any proposed proof for a No-instance will be rejected with probability at
least = 1/poly(N ). This rather weak requirement on the bare false-Acceptance rate
can be bootstrapped into an arbitrarily accurate verification scheme by repetition, as
sketched in Sec. 2.4 above.
Given an instance H =
P
Π, the obvious proof is for Merlin to provide a state
|Ψi which he alleges is a zero energy state. Arthur’s verification procedure will be to
check this claim. The verifier works by measuring each of the Π in some pre-specified
order on the state. That this can be done efficiently follows from the fact that Π
acts on no more than k qubits and therefore its measurement can be encoded in a
N -independent number of quantum gates. Clearly, if |Ψi is a zero energy state, it is
a zero-energy eigenstate of each of the Π and therefore all of these measurements will
produce 0 and the verifier accepts. This checks condition (a) above and we say our
verification scheme is complete.7
On the other hand, if H is a No-instance, it has a ground state energy above the
promise gap ∆ and |Ψi necessarily has overlap with the positive eigenspaces of at
least some of the Π. It is a short computation to show that the probability that all of
the measurements return 0 will then be bounded above by 1−∆/N k ∼ 1−1/poly(N ).
Thus, No-instances will be rejected with probability at least = 1/poly(N ) and our
verification scheme is sound.
7
The feature that one can do these measurements by local operations and that they provide
probability 1 verification of ground states is a special feature of the QSAT Hamiltonian which allows
it to somewhat evade the heuristic expectations of time-energy uncertainty.
21
|σ0 σ1 · · · σN i
|00 · · · 0i
U3
Accept
Proof
Ancilla
U1
U2
UT
t
0
1
2
T
Figure 2.3: QMA verification circuit. The circuit depends on the instance and must
be constructible by an efficient algorithm given the instance. We have drawn the
circuit so that there is a single local gate per time step, so T = poly(N ).
2.6.2
QSAT is QMA1 -complete
Just as in the classical Cook-Levin proof, we need to show that any QMA1 problem
can be reduced to solving an instance of QSAT. We again exploit the only thing
that all QMA1 problems have in common: their quantum verification algorithm.
We will take the quantum circuit representing this verifier and construct from it a
QSAT Hamiltonian whose ground state energy will be directly related to the maximal
acceptance probability of the verification circuit.
Let A be an arbitrary instance of a QMA1 problem L. Then A has a polynomial
sized quantum verification circuit as in Fig. 2.3. This circuit takes as input a quantum
state encoding a proof that A is a Yes-instance of L, along with some ancilla work
qubits in a fiducial 0 state, performs a sequence of T unitary one and two qubit
gates and then measures the output state of some subset of the qubits. If A is a
Yes-instance, then there exists a valid input state such that all of the output bits will
22
yield 0 with probability 1. Otherwise, at least one of the output bits will read 1 with
probability polynomially bounded away from 0 for any input state |σ0 · · · σN i.8
We now construct a single particle hopping Hamiltonian whose ground state encodes the operation of this quantum circuit. We introduce a clock particle hopping
on a chain of length T and label its position states by |ti. We endow this particle
with an enormous ‘spin’: a full N qubit Hilbert space (dimension 2N ). As the particle
hops, ‘spin-orbit’ coupling induces rotations on the N qubit space which correspond
to the unitary gates in the verification circuit. To wit:
Hp =
X1
t
2
†
(−|t + 1iht| ⊗ Ut+1 − |ti ht + 1| ⊗ Ut+1
+ |tiht| + |t + 1iht + 1|)
(2.2)
The terms of this Hamiltonian have been normalized and shifted such that each
is a projector with energies 0 and 1, but otherwise it is just a 1-D hopping problem
with Neumann boundary conditions. Indeed, there is a simple gauge transformation
in which the spin-orbit coupling disappears entirely. This consists of rotating the
basis of the position t spin space by U1† U2† · · · Ut† . In this representation, we see that
the 2N spin components decouple and the system is really 2N copies of the Neumann
chain. Thus, the spectrum has a cosine dispersion with bandwidth 1, ground state
energy 0 at wave vector k = 0 and allowed k = nπ/(T + 1).
The propagation Hamiltonian has (zero energy) ground states of the form (in the
original gauge):
X
1
|ti ⊗ Ut Ut−1 · · · U1 |ξi
|ψi = √
T +1 t
(2.3)
where |ξi is an arbitrary ‘input state’ for the N qubit space. This state is called the
history state of the computation performed by the verification circuit given input |ξi.
It is a sum over the state of the quantum computation at each step in the circuit
8
The thoughtful reader will notice the addition ancillae qubits. These are necessary when computation is done by reversible gates, as in unitary circuit computation. We leave it as an exercise to
the reader to figure out why the absence of ancillae would make the verification circuit unsound.
23
evolution. Any correct computation corresponds to a zero energy history state –
incorrect computations will overlap higher energy hopping states.
Now, we have a Hamiltonian that encodes the correct operation of the verification
circuit. We simply need to add terms that will penalize computations which do not
correspond to appropriate input states and accepting output states. These terms
effect the computational state at times t = 0 and T , so they are boundary fields
from the point of view of the hopping problem. In general, they should split the 2N
degeneracy of the pure hopping in Hp , and since they are positive operators, lift the
ground state energy.
The initialization term is simply a sum over the ancilla qubits of projectors penalizing |ξi with incorrectly zeroed ancillae:
Hi =
X
j∈Ancilla
|0ih0|t ⊗ |1ih1|j
(2.4)
Similarly, the output term penalizes output states which overlap |1i on the measured
output bits:
Ho =
X
j∈Accept
|T ihT |t ⊗ |1ih1|j
(2.5)
Now we consider the full Hamiltonian
H = Hi + Hp + Ho
(2.6)
If |ψi is a zero energy state of H, then it is a zero energy state of each of the three
pieces. Hence, it will be a history state in which the input state |ξi has appropriately
zeroed ancillae and the output state has no overlap with |1i on the measured qubits
– thus, |ξi is a proof that the verifier accepts with probability 1 and the original
instance A is a Yes-instance. Conversely, if such a proof state |ξi exists then the
history state built from it will have zero energy.
24
It is somewhat more work to show the soundness of the construction: A is a
No-instance if and only if the Hamiltonian H has ground state energy bounded polynomially away from 0 [22, 4]. The intuition is straightforward – the strength of the
boundary fields in the gauge-transformed hopping problem for a given spin state corresponds to the acceptance probability of the associated input state. Since these
repulsive fields are lifting Neumann conditions, this raises the ground state energy
quadratically in 1/T and the field strength. For a No-instance, no spin sector is both
valid and accepting, thereby avoiding both field effects.9
To be a bit more precise,10 we assume for contradiction that we have a state |ψi
with energy exponentially small in N (hence smaller than any polynomial in N or
T ):
hψ| H |ψi = hψ| Hi |ψi + hψ| Hp |ψi + hψ| Ho |ψi ≤ O(e−N )
(2.7)
Since each term is positive, each is bounded by the exponential. The hopping Hamiltonian Hp has a gap of order 1/T 2 for chains of length T , thus if we decompose |ψi
into a zero energy piece (a history state) and an orthogonal complement,
√
1 − α2 X
|ψi = √
|ti Ut · · · U1 |ξi + α |Exci
T +1 t
(2.8)
we must have exponentially small overlap onto the complement:
O(e−N ) > hψ| Hp |ψi = α2 hExc| Hp |Exci > α2 O(1/T 2 )
(2.9)
In other words,
|ψi = √
9
X
1
|ti Ut · · · U1 |ξi + O(e−N )
T +1 t
(2.10)
This is overly simplified: in the absence of the output term Ho , the gauge transformed problem
can be thought of as 2N decoupled hopping chains, some fraction of which have boundary fields at
t = 0. The output term is not simply a field on these chains – it couples them and in principle
allows hopping between them as a star of chains. The upshot is that the repulsive (diagonal) piece
outweighs the off-diagonal mixing.
10
Although still not fully rigorous.
25
The input term Hi has energy 0 for valid input states ξ V and energy at least 1 for
p
invalid states ξ I . Thus, decomposing |ξi = 1 − β 2 ξ V + β ξ I , we find (abusing
notation and dropping explicit reference to the |t = 0i sector on which Hi acts):
−N
O(e
β 2 I I β2
−N
) > hψ| Hi |ψi =
ξ Hi ξ + O(e ) ≥
+ O(e−N )
T +1
T +1
(2.11)
In other words,
|ψi = √
X
1
|ti Ut · · · U1 ξ V + O(e−N )
T +1 t
(2.12)
Finally, considering the output term we find:
hψ| Ho |ψi =
1 V †
pr
ξ U1 · · · Ut† Ho Ut · · · U1 ξ V + O(e−N ) =
+ O(e−N )
T +1
T +1
(2.13)
where pr is the probability that the original QMA1 verifier rejects the proposed proof
V
ξ
for the No-instance A. Since this rejection probability is bounded below by a
constant, the state |Ψi cannot possibly have exponentially small energy.
We have reduced the arbitrary QMA1 instance A to asking about the zero energy
states of a hopping Hamiltonian H constructed out of projectors. This is almost what
we want. We have a Hamiltonian constructed out of a sum of projectors but they
each act on three qubits tensored with a (large) particle hopping space rather than
on a small collection of qubits.
The final step in the reduction to k-QSAT is to represent the single particle Hilbert
space in terms of a single excitation space for a chain of clock qubits in such a way that
we guarantee the single particle sector is described by H above and that it remains
the low energy sector. We refer the interested reader to the literature [22] for more
details on these clock constructions. Each of the projectors of H becomes a joint
projector on one or two of the computational (spin) qubits and some number of the
26
clock qubits (two in [22]). The final 4-QSAT Hamiltonian will then be given by a
sum of projectors involving at most 4 qubits
H = Hi + Hp + Ho + Hc
(2.14)
where Hc acts on the clock qubits to penalize states which have more than one clock
particle.
2.7
Typical versus worst-case complexity
k-SAT is NP-complete for k ≥ 3. Thus, for any given algorithm, we expect that
there are instances which will take an exponentially long time to solve. However, we
ought not be too discouraged – some instances of k-SAT may be parametrically easier
to solve than others, and these may be the ones of interest in a given context. To
make this more precise, it is useful to introduce the concept of typical, as opposed to
worst-case, complexity.
In order to define typicality, one introduces an ensemble of problem instances
which assigns to each possible instance a probability of arising. Typical quantities are
then given by statistical statements, e.g. about the median time required for solving
problem instances. This may differ substantially from the corresponding worst-case
quantities when the latter have a sufficiently small weight in the ensemble. Precisely
what quantities to calculate depends on the aspects of interest. For instance, a median
run-time is not much affected by a small fraction of exponentially long runs, while
these may dominate the expectation value.
It is worth emphasizing that the polynomial reductions discussed in Sec. 2.2 provide a characterization of the worst case difficulty of solving problems. Reductions
and algorithms in this context must work for all instances of a problem – it may
be that typical instances are far easier to solve than the general case. Reductions
27
however may transform typical instances of A into rather atypical instances of B,
which are somehow less tractable than their typical counterparts. Whether a useful
framework of reductions can be defined that preserve typicality is an open question
(see Chapter 22 of [11]), but the study of typical instances of particular hard problems
has itself been a fruitful activity which will occupy us for much of the rest of this
thesis.
2.8
Classical statistical mechanics of k-SAT
We now give an account of an analysis of such an ensemble for classical k-SAT
[31, 50, 68, 91, 87]. For completeness, let us begin with reviewing the original definition of classical k-SAT, expanding somewhat on the brief definition provided in
Sec. 2.5 above. Indeed, the original computer science definition of satisfiability looks
somewhat different from the Hamiltonian problem we introduced. Consider a set of
N Boolean variables {xi | i = 1 . . . N }, i.e. each variable xi can take two values, true
or false (in which cases the negation x̄i is false or true, respectively). Classical k-SAT
asks the question, “Does the Boolean expression:
C=
M
^
Cm
(2.15)
m=1
evaluate to true for some assignment of the xi ?” Here, each clause is composed of a
disjunction of k literals, e.g. for k = 3:
Cm = xm1 ∨ x̄m2 ∨ xm3
(2.16)
where each variable occurs either affirmed (xm1 ) or negated (x̄m2 ). Hence, there are
2k possible clauses for a given k-tuplet {xmj | j = 1 . . . k}.11
11
By the way, the funny symbols ∧ and ∨ are the Boolean operators ‘and’ and ‘or’.
28
This definition is equivalent to the definition in terms of the k-body interacting
spin Hamiltonian of Eq. (2.1). To obtain a spin Hamiltonian from the collection of
clauses, Eq. (2.15), we convert the Boolean variables xi into Ising spins σi = ±1,
with σi = +1(−1) representing xi being true (false). A clause then becomes a k-spin
interaction designed such that the satisfying assignments evaluate to energy 0, and
the forbidden assignment to energy 1. For instance, the clause given in Eq. (2.16)
turns into:
Hm = 2−k 1 − σim1
1 + σim2
1 − σ im 3
The k-SAT ensemble is now random in two ways:
(R1) each k-tuple occurs in H with probability p = αN
N
k
.
(2.17)
(R2) each k-tuple occurring in H is randomly assigned one of the 2k possible clauses.
Here, we have introduced a parameter α, the clause density, which allows us to tune
the ensemble through various regimes. We note that the the number of clauses,
M = αN , is indeed proportional to the number of variables12 because there are Nk
possible k-tuples. The Hamiltonian can be pictorially represented by an interaction
graph, Fig. 2.4. This is a bipartite graph, one sublattice of which has N nodes, denoted by circles representing the xi , and the M nodes of the other sublattice denoted
by squares represent the clauses Cm . Each square is connected to the k variables
participating in the clause it represents, whereas each variable is connected to all
clauses it participates in, which implies an average coordination of αk, with a Poissonian distribution. The random graph thus constructed contains all the information
on a given problem instance if we label each square with which of the 2k possible
clauses it represents. This graph will be used for random quantum k-SAT as well,
where the Boolean variables and clauses will be replaced by appropriate quantum
generalizations (see Chap. 3).
12
Actually, only the expectation value of M equals αN . The Poissonian distribution for M of
course has vanishing relative fluctuations (hM2 i − hMi2 )/hM i2 as N → ∞.
29
Figure 2.4: Examples of interaction graphs for (a) 2-SAT and (b) 3-SAT, respectively.
The (green) circles represent qubits. (a) The clusters, clockwise from bottom left, are
chain, tree, clusters with one and two closed loops (“figure eight”). The short closed
loops, as well as the planarity of the graphs, are not representative of the thermodynamic limit of the random graph ensemble. (b) Each (blue) square represents a clause
connected to 3 nodes. Clockwise from top left are a tree, a graph with nontrivial core
and a graph with simple loops but no core.
We now note a few key properties of the interaction graph ensemble: At small
clause density α, the size distribution of connected components (“clusters”) is exponential. Moreover, almost all of these clusters are treelike with a few, O(N 0 ),
containing a single closed loop (see Fig. 2.4); the number of clusters with several
closed loops vanishes as N → ∞. Above a critical value αgc (k) =
1
,
k(k−1)
a giant
component emerges on which a finite fraction of the qubits reside. Unlike the finite clusters, the giant component may contain a non-vanishing core, defined as the
subgraph remaining after recursively stripping away leaf nodes (i.e. nodes of degree
1). For k = 2, an extensive core emerges continuously at αhc (2) = αgc (2) = 12 ; for
k ≥ 3, the core appears abruptly with a finite fraction of the nodes at αhc (k) > αgc (k)
[92, 95].
30
αd+
αd
αc
αs
Figure 2.5: Schematic phase diagram for random classical constraint satisfaction
problems such as random k-SAT. Black dots represent connected clusters of solutions
in configuration space and the sequence of transitions reflect various rearrangements
of the clusters. Figure after [72].
2.9
Phase diagram of classical random k-SAT
In Fig. 2.5, we show a schematic phase diagram for random k-SAT. The first question
one might ask is: is there a well-defined phase transition, at some value α = αs (k),
such that instances for α < αs are satisfiable, and those for α > αs are not? It has
been shown that there exists such a transition for the random ensemble. This does
not mean that all instances with α < αs are satisfiable: given an UNSAT instance
with N sites and αN clauses, one could simply add N disconnected sites to get a
new UNSAT instance with α0 = α/2. What is true instead is that the probability of
having such an UNSAT graph with α0 < αs is exponentially small in N , so that for
N → ∞, such graphs do not arise with a probability approaching 1.
It is easy to provide a very rough estimate for where this happens, by adapting
an idea of Pauling’s which was originally devised for estimating the configurational
entropy of the protons in water ice. We consider the clauses as constraints, each of
which ‘decimates’ the number of allowed configurations by a factor (1 − 2−k ): only
1 out of the 2k possible configurations of variables of any given clause is ruled out.
For M = αN such constraints, one is left with 2N (1 − 2−k )αN solutions. In the
thermodynamic limit, this number vanishes for α > αwb = −1/ log2 (1 − 2−k ) ∼ 2k .
In the k-SAT literature, this is known as the ‘first-moment bound’, for which there is
a straightforward rigorous derivation.
31
The SAT-UNSAT transition is not the only transition of this problem, though. As
indicated in Fig. 2.5, statistical mechanical methods imported from the study of spin
glasses have been used to establish finer structure in the SAT phase. This plot shows
a set of cartoons of configuration space, indicating the location of satisfying assignments. For N variables, configuration space is an N -dimensional hypercube and this
plot indicates, in a two-dimensional ‘projection’, how ‘close’ satisfying assignments
are to each other. Roughly, two solutions belong to the same cluster if they can be
reached via a sequence of satisfying configurations such that two consecutive ones
differ by O(N β ) variables with β < 1 [72].
Figure 2.5 thus documents a set of transitions in the clustering of satisfying assignments. For the smallest α, all solutions belong to one single giant cluster – the
full hypercube for α = 0 – and then there is a successive break-up into smaller, and
increasingly numerous clusters as α grows [72].
This structure of configuration space should have ramifications for how hard it
is to solve the corresponding instances: small-scale clusters indicate a rugged energy
landscape, with numerous local minima providing opportunities for search algorithms
to get stuck. Indeed, all known algorithms slow down near αs critical. That said,
many simple approaches to random k-SAT problems actually do quite well even in
the clustered phases and the detailed relationship between clustering in configuration
space and algorithmic difficulty is subtle, somewhat detail dependent and an ongoing
research topic [125].
The derivation of this phase diagram was obtained using methods imported from
the study of spin glasses, in particular the cavity method [72, 60]. The insights thus
gained have lead to the development of an impressive arsenal of techniques for not
only determining whether or not a k-SAT problem instance is soluble, but also for
actually finding solutions in the form of satisfying assignments [21, 86, 93, 91]. We
will have much more to say regarding the cavity method in Part II of this thesis and
32
refer the impatient reader to Chapter 4 for a more detailed introduction to these
techniques.
2.10
Adiabatic quantum algorithm for classical kSAT
Before we move on to the generalized problem of quantum satisfiability k-QSAT,
let us first ask whether we could use a quantum algorithm to solve classical k-SAT
efficiently. One possible strategy for this is based on a protocol known as adiabatic
quantum computing [43, 69], which in its full generality is equivalent to computation
based on circuits. Here we will discuss a particularly simple member of this class of
algorithms.
Consider a time-dependent quantum Hamiltonian, with real time t parametrized
by s(t) (with 0 ≤ s ≤ 1):
H(s) = (1 − s)HΓ + sH0
,
(2.18)
where HΓ is a transverse-field term and H0 is obtained from Eq. (2.1)
X
HΓ = −Γ
σix
X
H0 =
Em ({σiz })
,
(2.19)
m
by replacing the σi ’s by Pauli operators σiz .
The ground state space of H(s) at time s = 1 is spanned by all lowest-energy
classical configurations (ground states) of the k-SAT problem; it is these which we
are after, but which can be hard to find. By contrast, at time s = 0, the quantum
ground state has all spins polarized in the x-direction. This is both easy to describe
33
and to prepare. If we start the system in its ground state at s = 0, and change H(s)
sufficiently slowly, the state of the system will evolve adiabatically, and reach the
desired state at time s = 1.
However, we do not want to change H(s) arbitrarily slowly, as this would be no
gain over an exponentially long classical run-time.
What is it that limits the sweep-rate? A non-zero sweep rate can induce transitions to excited states. Careful derivation of the adiabatic theorem reveals that
the probability of nonadiabatic transitions tends to zero so long as the sweep rate is
slower than the minimal adiabatic gap ∆ squared13 . That is, the run time T must be
greater than or of order O(1/∆2 ) in order to ensure adiabaticity.
Heuristically, we need to be concerned about avoided level crossings in the course
of the evolution and in particular, the avoided crossing at the location of the minimal
gap ∆.14 As two levels approach closely, we get an effective two-level problem:

 α(t − t0 )
H2 = 
∆/2
∆/2
−α(t − t0 )



.
(2.20)
At the closest approach, at t = t0 , the ground state is separated from the excited
state by a gap ∆. For |α(t − t0 )| ∆, variation of t has very little effect on the
adiabatic eigenstates and the Schrödinger evolution remains adiabatic even for fast
sweeping. It is only the time spent in the interaction region |α(t − t0 )| < ∆ where the
adiabatic states rotate significantly and nonadiabatic transitions may arise. Thus,
the interaction time is tI ∼ ∆/α and the dimensionless figure of merit for adiabatic
behavior should be ∆ · tI ∼ ∆2 /α. In particular, for low sweep rates α ∆2 or
long run times T ≥ O(1/∆2 ), we expect to have purely adiabatic evolution. We note
13
In fact, there is some controversy in the rigorous literature about whether the asymptotic sweep
rate must be slower than 1/∆2 or 1/∆2+δ for some arbitrarily small constant δ. See [5].
14
In the absence of any symmetries and fine-tuning, all level crossings are avoided as a function
of a single adiabatic parameter s.
34
that the nonadiabatic transition probability P for this two-level model was calculated
some eighty years ago by Landau [76] and Zener [126] whose exact result:
P = 1 − e−π∆
2 /4~α
(2.21)
quantifies the physical intuition in this case.
The quantum adiabatic algorithm has been studied extensively since its introduction ten years ago on a number of hard random constraint satisfaction problems
closely related to 3-SAT [112, 43]. The critical question is simple: how does the
typical minimal gap encountered during the procedure scale with increasing instance
size N ? Analytic work on simple (classically easy) problem ensembles found several
polynomial time quantum adiabatic algorithms. Moreover, early numerical studies
for very small instances of harder problems held promise that the gap would scale
only polynomially [121, 43]. Unfortunately, subsequent numerical studies on larger
systems indicate that the gap eventually becomes exponentially small due to a first
order transition between a quantum paramagnet and a glassy state [122]. Even worse,
recent perturbative work argues that many-body localization leads to a plethora of
exponentially narrow avoided crossings throughout the glassy phase [6, 7] and thus
the algorithm discussed here does not produce an efficient solution of random 3-SAT
or related random constraint satisfaction problems.
35
Chapter 3
Random Quantum Satisfiability
3.1
Introduction
Let us now turn to the study of random instances of quantum satisfiability k-QSAT1 .
As discussed in Sec. 2.6, k-QSAT is QMA1 -complete and thus should be generally
intractable. As in the classical case, one might hope to gain some insight into the
nature of the difficulty of the quantum problem by studying a random ensemble of
its instances. Moreover, the richness of phenomena exhibited by the classical random
satisfiability problem – and the many important spin-off techniques that have been
developed in their study – encourages us to seek analogous behaviors hiding in the
quantum system.
Let us recap the definition of k-QSAT from Sec. 2.6:
Input: A quantum Hamiltonian H =
P
m
Πm composed of M projectors, each acting
on at most k qubits of an N qubit Hilbert space.
Output: Does H have a zero energy ground state?
Promise: Either H has a zero energy state or all states have energy above a promise
gap ∆ > 1/poly(N ).
1
Chapter based on work with A. Läuchli, R. Moessner, A. Scardicchio and S. L. Sondhi [78, 77].
36
Quantum satisfiability is a natural generalization of the classical satisfiability problem:
bits become qubits and k-body clauses become k-body projectors. In key contrast
to the classical case, where the binary variables and clauses take on discrete values,
their quantum generalizations are continuous: the states of a qubit live in Hilbert
space, which allows for linear combinations of |0i and |1i. Thinking of a Boolean
clause as forbidding one out of 2k configurations leads to its quantum generalization
as a projector ΠIφ ≡ |φihφ|, which penalizes any overlap of a state |ψi of the k qubits
in set I with a state |φi in their 2k dimensional Hilbert space. Indeed, if we restrict
the Πm to project onto computational basis states, k-QSAT reduces back to k-SAT
– all energy terms can be written as discrete 0 or 1 functions of the basis state labels
and the promise gap is automatically satisfied since all energies are integers.
As in the classical problem, we make two random choices in order to specify an
instance:
(R1) each k-tuple occurs in H with probability p = αN
N
k
(R2) each k-tuple occurring in H is assigned a projector Πm = |φihφ|, uniformly
chosen from the space of projectors of rank r. For these notes, we will mostly
consider the case r = 1, although higher rank ensembles can be studied [101].
The first rule is identical to that of the classical random ensemble and thus the
geometry of the interaction graphs (Fig. 2.4) is the same – locally tree-like with long
loops for sufficiently high clause density α.
The second rule, however, dramatically changes the nature of our random ensemble: the measure on instances is now continuous rather than discrete. Allowing
the choice to range uniformly over the qubit Hilbert space has the paradoxical effect
of including non-classical instances and simultaneously making many features of the
QSAT ensemble hold generically, or with probability 1, with respect to this choice.
We will return to this important observation in Sec. 3.3.1. Throughout the rest of
37
0.17
0.81 0.92
αgc
αhc αps
1/2
1
αgc = αps
αcl
k=2
3.59 4.26
αsf
7.49
αcl
k=3
αf m
0
+
1-
2k
12e k 2
ln 2 k
2
2
ln 2 2k
αhc
αps
αqlll
αnose
αcl
PRODSAT
SAT-UNPRODSAT
k→∞
UNSAT
Figure 3.1: Phase diagram of k-QSAT as a function of clause density α at k = 2,
k = 3 and as k becomes large.
this chapter, we use the term generic to refer to the continuous choice of projectors
and random to refer to the choice of the graph.
3.1.1
Phase diagram
The first step in understanding the random ensemble is to compute the statistics
of this decision problem as a function of α. Specifically we would like to know if
there are phase transitions in the satisfying manifold as α is varied: these include
both the basic SAT-UNSAT transition as well as any transitions reflecting changes
in the structure of the satisfying state manifold. Additionally, we would like to check
that the statistics in the large N limit are dominated by instances that automatically
satisfy the promise gap.
The current state-of-the-art QSAT phase diagram is in figure 3.1. Let us walk
through a few of the features indicated. First, we have separated the k = 2 case from
the higher connectivity cases because it is significantly simpler. For k = 2, we can
solve the satisfiability phase diagram rigorously and even estimate energy exponents
for the (non-zero) ground state energy above the satisfiability transition. Our ability
to do so is consistent with the fact that 2-QSAT is in P – instances of 2-QSAT can be
efficiently decided by classical computers! In particular, it can be shown that the zero
38
energy subspace can be spanned, if it is nontrivial, by product states. Since these are
much simpler to specify classically (a product state needs only 2N complex numbers
instead of 2N ), it is perhaps not surprising that we can decide whether or not such
states exist that satisfy a given instance H. In any event, the only significant feature
of the phase diagram is that for α < αs = 1/2, we have a PRODSAT phase – that is
a phase which is satisfiable by unentangled product states – and for α > αs , we have
an UNSAT phase with finite ground state energy density. The transition coincides
with a geometric transition: αs = 1/2 corresponds to the emergence of the a giant
component in the underlying interaction graph. We will derive all of these results in
more detail in Sec. 3.2.
For k ≥ 3, the phase diagram is somewhat more interesting. Again, at low
α < αps (k) ∼ 1 there exists a PRODSAT regime in which satisfying product states
are guaranteed to exist. Above αps , there are no satisfying product states, but the
system remains SAT – thus, there is an entanglement transition in the ground state
space as a function of α. Finally, above some αc ∼ 2k , there is an UNSAT phase in
which it can be shown that there are no zero energy satisfying states. We note that
the emergence of a giant component in the underlying interaction graph happens at
αgc =
1
k(k−1)
αps αc – the various relevant transitions are well-separated at large
k. A variety of different techniques go in to showing the existence of these transitions
and phases – we will sketch most of these arguments in Sec. 3.4.
3.2
2-QSAT
Let us first consider our questions in the specific context of 2-QSAT (phase diagram
in Fig. 3.2). While this is a classically easy (P) problem, the random ensemble has
much of the structure we also find for the harder k > 2 cases. An instance H of
QSAT is satisfiable if its kernel ker(H) has nonzero dimension; we call this dimension
39
the rank RH of the QSAT instance H.2 In the following, we will find the kernel
of H explicitly by transforming the random projector problem almost surely into a
Heisenberg ferromagnet, whose ground state space is well known.
A central tool in this analysis is Bravyi’s transfer matrix Tφij which, given a vector
|ψi i of the state of qubit i yields |ψj i = Tφij |ψi i for qubit j, such that the product
state |ψij i = |ψi i ⊗ |ψj i satisfies the projector onto φ: hψij |Πij
φ |ψij i = 0. Concretely,
if the projector penalizes a joint state of both qubits given by the complex vector
φ = (φ00 , φ01 , φ10 , φ11 ), we have
†
Tφ = φ =
+φ∗01 + φ∗11
−φ∗00 − φ∗10
.
(3.1)
Here is the standard antisymmetric matrix (Levi-Civita symbol) in two dimensions.
We note that the transfer matrix T ij for any given link is almost surely invertible so
this construction finds satisfying product states for any input |ψi i.
3.2.1
Trees are SAT
Consider the clusters that enter the 2-QSAT graph ensemble. Of these, a tree comn
N
prising n qubits and n − 1 edges has a satisfying product state |Ψi =
|ψj i, where
j=1
|ψj i is obtained from an arbitrary reference qubit i = 1 by repeated application of
the T ’s along the (unique) path joining i with j. In fact, the satisfying subspace is,
almost always, n + 1 dimensional. To show this, we will map the random projector problem directly onto the ferromagnetic Hamiltonian on the same tree. In more
mathematical terms, we construct a non-unitary action of the permutation group on
the qubit Hilbert space that leaves the zero energy space ker(H) invariant. Thus,
2
We note that this terminology is somewhat confusing since one usually thinks of the rank of a
linear map H as the dimension of the image of H, rather than the dimension of its kernel (often
called the nullity). This confusion arises due to the alternative formulation of QSAT in terms of the
intersection of the eigenvalue 1 spaces of inverted clause projectors 1 − Π. We hope that this double
use of rank will not pose too great a strain on the reader.
40
the ground state space is precisely the totally symmetric space Symn C2 , which has
dimension n + 1.
We define a particular, non-orthogonal product basis for the Hilbert space H =
H0 ⊗ H1 ⊗ · · · ⊗ Hn−1 . First, choose any two linearly independent unit vectors |↑0 i
and |↓0 i as a basis for H0 . Second, use the transfer matrix T 0j to transfer this basis
of H0 to a normalized basis
0j
j
j
↑ = T |↑ i ,
kT 1j |↑j ik
0j
j
j
↓ = T |↓ i
kT 1j |↓j ik
(3.2)
of Hj for each of the neighbors j of 0. Finally, recursively traverse the tree to produce
a pair of vectors |↑i i , |↓i i for each site i in the tree. This procedures produces a choice
of basis for each of the individual qubit Hilbert spaces which we use to define a product
basis we call the transfer basis. Although we use the notation of spin up and down,
we emphasize that the states need not be orthogonal.
Finally, we show that the ground state space of the Hamiltonian H is precisely
the space of totally symmetric states defined with respect to the transfer basis. We
consider the constraint that a generic vector Ψ ∈ H is annihilated by Π01 by factoring
Ψ:
|Ψi = ↑0 ↑1 v12...n−1 + ↓0 ↓1 v22...n−1 + ↑0 ↓1 + ↓0 ↑1 v32...n−1
+ ↑0 ↓1 − ↓0 ↑1 v42...n−1 .
(3.3)
The first three terms are identically annihilated by the projector Π01 . This follows
immediately from the construction of the transfer basis for the first two terms while
41
for the third term (summing over repeated indices):
Π01 ↑0 ↓1 + ↓0 ↑1
= φ01 (φ01∗ )α0 ,α1 ↑0α0 ↓1α1 +(φ01∗ )α0 ,α1 ↓0α0 ↑1α1
= φ01 ↓0T φ01∗ φ01† ↑0 + ↑0T φ01∗ φ01† ↓0
= 0
(3.4)
where we have exploited the antisymmetry of . Thus, |v4 i must be zero and Ψ clearly
lies in the symmetric eigenspace for swaps (01). A similar argument holds for each
of the swaps (ij) on links of the tree B. Since B is connected, these swaps generate
the full permutation group Sn and Ψ must be in the completely symmetric subspace
of its action. In particular, this means that the ground state space is isomorphic to
Symn C2 which is n + 1 dimensional.
3.2.2
Loops
To understand hypergraphs with loops, we first look for satisfying product states. For
a cluster G comprising nL independent closed loops (i.e. n qubits with n + nL − 1
edges, see Fig. 2.4), a product state is internally consistent only if the product of the
T ’s around each closed loop returns the state it started with. For a graph with a
single closed loop, O1 , this imposes
Y
jk∈O1
T jk
!
|ψi i = λ1 |ψi i,
(3.5)
which in general allows two solutions, so that these graphs are SAT. By contrast, if
a site i is part of a second independent closed loop, O2 , the additional demand
Y
jk∈O2
T jk
!
|ψi i = λ2 |ψi i
42
(3.6)
already yields an overconstraint. Thus graphs with more than one closed loop lack
satisfying product states with probability 1.
In fact, we can generalize the above reasoning to show that it holds even for the
entangled (non-product) states in the problem as follows. We choose a spanning
tree on G and starting qubit 0 to define a transfer basis – the ground state space is
necessarily a subspace of the associated symmetric space. In particular, if G contains
a single closed loop O1 , we take qubit 0 on the loop. In this case, we choose the
starting basis on qubit 0 to satisfy the product state consistency condition for O1
Y
m∈O1
!
T m ψ 0 = λ ψ 0 ,
(3.7)
i.e. we take the eigenbasis of the loop transfer matrix, which we denote ↑0 , ↓0 with
eigenvalues λ↑ , λ↓ . These two states transfer to two linearly independent product
ground states for the loop – all up and all down. A short calculation verifies that in
fact there are no more: write a generic Ψ ∈ ker(H) ⊂ Symn C2 as
|Ψi = ↑−1 ↑0 |v1 i + ↓−1 ↓0 |v2 i + ↑−1 ↓0 + ↓−1 ↑0 |v3 i ,
(3.8)
where −1 is the qubit at the end of the loop O1 and vi are vectors on the n − 2 other
qubits. The projection Πh−1,0i annihilates the first two terms by choice of the basis.
However:
Πh−1,0i |Ψi = φh−1,0i (↑−1T φh−1,0i† ↓0 + ↓−1T φh−1,0i† ↑0 ) |v3 i
= φh−1,0i (λ↑ ↑0T ↓0 +λ↓ ↓0T ↑0 ) |v3 i
= φh−1,0i (λ↑ − λ↓ )(↑0T ↓0 ) |v3 i
(3.9)
Since λ↑ 6= λ↓ w.p. 1, v3 must be zero and by symmetry, ψ must be in the span of the
all up and all down states.
43
ǫ0 = E0 /N
O(α − αq )2
SAT
O(α − αc )3
UNSAT
αq =
1
2
αc = 1
α
Figure 3.2: Phase diagram of 2-QSAT. The quantum SAT-UNSAT transition αq = 12
coincides with the emergence of a giant component in the random graph, which lies
at half the classical 2-SAT transition αc = 1 [31, 96]. The solid (dashed) line marks
an asymptotic upper bound on the quantum (classical) ground state energy density
0 .
An analogous calculation verifies that any additional closed loops introduce projectors that are violated on this two-dimensional subspace and thus graphs with more
than one loop are unsatisfiable.
It is worthwhile to consider how the usual Heisenberg ferromagnet fits into the
above calculations. Up to scaling and an offset, this Hamiltonian is a 2-QSAT Hamiltonian whose terms all project onto the spin singlet |↑↓i − |↓↑i of adjacent spins. The
transfer matrices are all equal to the identity and thus all of the loop constraint conditions are identically satisfied. From the point of view of Eq. (3.9), the ferromagnet
has λ↑ = λ↓ = 1 and there is no reduction in the ground state degeneracy due to the
introduction of loop-closing ferromagnetic bonds.
3.2.3
Phase diagram
In light of the above analysis, the existence of a SAT/UNSAT phase reduces to the
presence of multiply connected components in the ensemble of 2-SAT graphs. For
α < αgc = 21 , the number of such clusters vanishes in the thermodynamic limit, so
any instance is SAT with probability that approaches 1. At αgc = 1/2, closed loops
proliferate as a giant component appears and thereafter all instances are UNSAT
with probability approaching 1. A straightforward upper bound on the energy in the
44
UNSAT phase, E ≤ O((α − αgc )2 N ), follows from the fact that the fraction of sites
in the core of the giant component grows as (α − αgc )2 .
Physical intuition suggests that the ground state energy should be extensive above
the transition and thus likely saturate the given upper bound. In the next section, we
adduce strong evidence for this by exhibiting a minimal lower bound on the energy
E > N 1− for any > 0 which holds except for exponentially rare instances. This
follows from the twin claims that (i) the energy of a figure eight comprising n sites
decays only polynomially with n, and (ii) that the number of disjoint figure eights
in the random graph grows nearly linearly with N . Observe that this kind of lower
bound is exactly what we need to establish that our ensemble keeps the promise that
either E = 0 or E > 1/N a with probability approaching 1. While this demonstration
is not strictly needed for 2-QSAT (since it is in P), it is suggestive of what we might
expect for the k ≥ 3 cases.
3.2.4
Extensivity and the promise gap
We expect on physical grounds that the UNSAT phase of k-QSAT has extensive
ground state energy with relatively vanishing fluctuations for any k. In this case,
the promise that E ≥ O(N −a ) fails to be satisfied only with polynomially small
probability by Chebyshev’s inequality. More generally, so long as the average ground
state energy is bounded below by a polynomially small scale E ≥ O(N −b ) with
relatively vanishing fluctuations, the promise will be violated with only polynomially
small probability for a > b.
Placing rigorous lower bounds on the expected quantum mechanical ground state
energy is generally difficult, but at least for k = 2, we argue as follows to find a
nearly extensive bound scaling as N 1− for any > 0. As the Hamiltonian for QSAT
is a sum of nonnegative terms, we can bound the ground state energy from below by
considering manageable subgraphs and ignoring the contribution from other terms.
45
In particular, the average ground state energy of a figure eight graph, that is a loop
of length L with one additional crossing edge, is polynomially bounded below by
O(L−δ ). This already gives a polynomial lower bound on the expected energy simply
by allowing L to scale as N and knowing that we will find at least one such subgraph
in the giant component with probability exponentially close to 1. We will do better
by finding a large number of disjoint such figure eight graphs.
The expected number of subgraphs A in the random graph GN,p is given by the
following formula:
E#(A ⊂ G) =
N!
pe(A)
(N − |A|)!|Aut(A)|
(3.10)
where |A| is the number of vertices in A, e(A) is the number of edges in A and
Aut(A) is the group of automorphisms of A (|Aut(A)| is the cardinality of this set).
This formula simply counts the number of ways of finding permutations of |A| nodes in
G and connecting them up into an A subgraph. For the fixed clause density ensemble,
we take p = 2α/N .
A figure eight graph is uniquely specified by giving its size L (we take L even) and
the distance d = 2 . . . L/2 between the two nodes that are connected by the crossing
link. We let A be the disjoint union of K figure eight graphs with L nodes each
and cross bars at separation L/2 − 1. Such a graph has |Aut(A)| = K!2K from the
K! permutations of the disjoint subgraphs and the two-fold symmetry of each figure
eight. Thus, the expected number of K-fold disjoint figure eights of size L is
E# =
N !pK(L+1)
(N − KL)!K!2K
46
(3.11)
We now allow K, L to scale with N such that KL N and use Stirling’s formula to
find the asymptotic entropy
KL
S ≈ KL log 2α −
+ K (log α + 1 − log K − log N ) .
N
(3.12)
This entropy is positive and growing with N so long as L log N , KL N and
α > αg = 1/2. In this regime, assuming that the fluctuations in S are relatively
small, we find that there are K ∼ N 1− disjoint figure eights of size L ∼ N /2 with
exponentially high probability. These lead to a nearly extensive lower bound on the
expected ground state energy.
3.3
Geometric Properties of QSAT
One of the most useful tools for studying random quantum satisfiability is geometrization. That is, the satisfiability of a generic instance of QSAT is a purely geometric
property of the underlying interaction graph. This point of view extends to many
properties of generic instances of QSAT – such as whether they are product satisfiable or not, etc. Results about generic QSAT thus follow from identifying the right
geometric properties to look for in the interaction graph ensemble. Indeed, this raises
the extremely interesting possibility that generic k-QSAT can be formulated without
reference to quantum Hamiltonians at all.
In this section we collect a set of rigorous results regarding geometrization and
generic product satisfiability that hold for any k (even mixed!), any rank (suitably
generalized) and any particular (non-random) interaction graph G. We will apply
these results to the general k random ensembles in Sec. 3.4.
47
3.3.1
Geometrization theorem
Geometrization Theorem. Given a fixed interaction graph G, the dimension RG =
| ker H| of the satisfying subspace of H is almost always minimal with respect to the
choice of projectors Πm . Thus, if G can be frustrated by some choice of projectors, it
is frustrated for almost all such choices.
PM
Proof. For a fixed interaction graph G with M edges, H = Hφ =
i=1 Πi =
PM
k
i=1 |φi i hφi | is a matrix valued function of the 2 M components of the set of M
vectors |φi i. In particular, its entries are polynomials in those components. Choose
|φi such that H has maximal (matrix) rank D. Then there exists an D × D submatrix of H such that det(H|D×D ) is nonzero. But this submatrix determinant is a
polynomial in the components of |φi and therefore is only zero on a submanifold of
the |φi of codimension at least 1. Hence, with probability 1, H has rank D and the
degeneracy | ker H| = 2N − D = RG .
We call RG the generic rank of the QSAT problem on a given interaction graph
G. The theorem holds for general rank r problems as well by a simple modification
of the argument to allow extra φ’s to be associated to each edge. Of course, RG also
depends on the ranks r of the projectors placed on G; since we will mostly focus on
r = 1 in this thesis, we suppress this dependence. For the purpose of calculating RG ,
it is also possible to view higher rank clauses on a graph G as rank 1 clauses on a
related graph G0 where each edge gets duplicated r times.
A nice corollary of this result is an immediate upper bound on the size of the
SAT phase. Consider any assignment of classical clauses on a given hypergraph: we
can also think of this as a special instance of k-QSAT where the projectors are all
diagonal in the computational basis. As this is a non-generic choice of projectors, the
dimension of its satisfying manifold is an upper bound on the dimension for generic
48
choices. We conclude then that the classical UNSAT threshold is an upper bound
on the quantum threshold. Indeed, if we can identify the most frustrated assignment
of classical clauses on a given hypergraph, i.e. the assignment that minimizes the
number of satisfying assignments, we could derive an even tighter bound. We note
that this classical problem, known as adversarial satisfiability (ADSAT) has been
studied [123, 109] through cavity analysis but so far has revealed no significant new
bounds.
Corollary 1. The rank of the random quantum problem is bounded above (w.p.1) by
the number of satisfying assignments of the most constrained classical k-SAT problem
on the same graph.
It is easy to construct example instances in which the quantum problem has fewer
ground states than the most frustrated classical problem on the same interaction
graph. Thus, the bound of Corollary 1 is not tight.
In the k = 2 case, Corollary 1 gives us another way to show the critical point
corresponds to the emergence of a giant component: once a giant component exists,
it will contain a giant loop with two crossing bonds. This graph can be made unsatisfiable by setting each of the clauses around the loop to disallow the state 01 and
the crossing bonds to disallow 00 and 11 respectively. Thus, the quantum problem is
generically UNSAT.
3.3.2
Product satisfiability
One of the remarkable results in Bravyi’s original work on QSAT was the explicit
construction of satisfying product states for k = 2, thanks to which 2-QSAT turned
out to be easy to decide and the concomitant satisfying states easy to construct. This
49
poses the question whether there exist regimes for k ≥ 3-QSAT in which analogous
results hold. We therefore define:
Definition 1. An instance of QSAT, (G, Πm ), is PRODSAT if it has a satisfying
product state. That is, ∃ |Ψi = |ψ1 i ⊗ · · · ⊗ |ψN i s.t. Πm |Ψi = 0 ∀m ∈ G.
PRODSAT may be viewed as a decision problem in its own right and, in the
presence of a polynomially small promise gap δ, it is efficiently verifiable by classical
computation. A witness is simply a collection of 2N C components of a satisfying
product state, the energy of which may be evaluated in linear time. Moreover, by
choosing Πm to project onto computational basis states, it is clear that PRODSAT
contains SAT and is therefore NP-complete.
In this section, we investigate product satisfiability for fixed graphs G and general
choices of rank 1 projectors Πm = |φm i hφm |. We show that G supports satisfying
product states on open neighborhoods in the projector manifold if it has a qubitclause dimer covering that covers all of the clauses – i.e.. an assignment of a unique
qubit to each clause (see Fig. 3.3). Appealing to the algebraic properties of the space
of projectors, this shows that G is in fact PRODSAT for all choices of Πm . It also
suggests a counting criterion for the ground state degeneracy. Thus, we will show
Theorem 2. If an interaction graph G, viewed as a bipartite factor graph of qubits
and clauses, has a dimer covering of its clauses, then
k −1
1. there exist open neighborhoods U ⊂ (CP2
)M of the rank 1 projector manifold
such that Πm ∈ U ⇒ (G, Πm ) is PRODSAT.
k −1
2. Moreover, (G, Πm ) is PRODSAT for all Πm ∈ (CP2
)M .
As a practical matter, product satisfiability is a question about the existence of
solutions to a collection of M homogeneous equations in N CP1 degrees of freedom
50
(a)
(b)
Figure 3.3: (a) Example of a k = 3 interaction graph with M < N . Circles (green)
indicate qubits and squares (red) indicate clause projectors that act on adjacent
qubits; (b) a dimer (shaded blue) covering that covers all clauses.
(unless otherwise noted, paired i indices contract, but m indices do not):
i1
i2
ik
φm
i1 i2 ···ik ψm1 ψm2 · · · ψmk = 0 ∀ m ∈ G
(3.13)
i1
where ψm
is the i1 ’th component of the state on the m1 ’th qubit and φm∗ is the
1
state onto which Πm projects. Since we are interested in generic choices of the φm ,
one might expect naive constraint counting to determine whether (M ≤ N ) or not
(M > N ) a given graph G is generically PRODSAT. Unfortunately, things are not
quite so simple because a G with a low average density (M ≤ N ) may still have
subgraphs that are dense enough to overconstrain the equations.
To make this last observation sharper, let us briefly restrict our attention to product projectors. In this case, Eq. (3.13) reduces to
m k ik
m 2 i2
i1
1
(φm
i1 ψm1 )(φi2 ψm2 ) · · · (φik ψmk ) = 0 ∀ m ∈ G
(3.14)
This equation is clearly satisfied if any of its factors is satisfied, which requires at least
one of the k qubits to be (uniquely) fixed orthogonal to the local projector. Thus, for
generic φ of product form, in which the local projectors on a site n are not parallel,
the full set of M equations is solvable if and only if one can uniquely associate a
qubit n to each clause m. Graphically, this is equivalent to the existence of a dimer
51
covering of the bipartite factor graph of G which covers all M clauses. Note that we
refer to matchings as dimer coverings if each clause is uniquely associated to a qubit
even if some qubits are left unpaired. See Fig 3.3.
We now relax the constraint that φ take product form and argue that the existence of the above dimer covering is still the relevant constraint counting for product
satisfiability. Let us consider an instance (G, φ) and assume that it is PRODSAT;
w.l.o.g. we may choose local bases such that |Ψi = 01 ⊗· · ·⊗ 01 is a satisfying assign
ment. Choosing stereographic coordinates z1n for each qubit, we find the constraint
equation is:
i1 i2
ik
φm
i1 i2 ···ik zm1 zm2 · · · zmk = 0
(3.15)
where now superscript i1 indicates exponentiation and the i indices run from 0 to 1.
Since zn = 0 satisfies this equation, we find that φm
00···0 = 0 necessarily but that the
remaining components of the φm are unconstrained.
Now we perturb φm 7→ φm + δφm and attempt to follow the solution zn = 0 7→
zn + δzn . To linear order, we have (for each clause m ∈ G):
m
m
m
φm
10···0 δzm1 + φ01···0 δzm2 + · · · + φ00···1 δzmk = −δφ00···0
(3.16)
This is a collection of M sparse linear inhomogeneous equations for the N variables
δzn . Let Ωmn be the M × N matrix of coefficients on the left hand side. This
matrix connects qubits to clauses and is sparse in the same pattern as the node-edge
adjacency matrix Amn of G.
Equation (3.16) is solvable for arbitrary δφm
00...0 if and only if the matrix Ω is
surjective. Trivially, this requires M ≤ N , but it also requires that some M ×
M subdeterminant of Ω be nonzero. Let us define a polynomial discriminant of
52
surjectivity as the tuple of all possible subdeterminants of Ω:
discτ (Ω) =
X
σ∈SM
(−1)σ Ω1τ (σ(1)) · · · ΩM τ (σ(M ))
(3.17)
where τ : M ,→ N runs over all injective maps. For formal components φ, discτ (Ω) is
not identically zero if and only if one of the terms in the above sums is not identically
zero; each non-zero term corresponds to a particular pairing of clauses and qubits in
the adjacency graph which covers all M clauses but need not cover all of the nodes.
Thus if G has such a dimer covering, Ω is almost always surjective, whereas it is never
surjective otherwise.
We can summarize the above analysis as follows: for G having dimer coverings, generic choices of φ for which (G, φ) is PRODSAT may be extended to open
neighborhoods around φ which are PRODSAT. Moreover, G has product states at
generic (non-parallel) product projectors. One can repeat the perturbative analysis of
Eq. (3.16) near such product projectors and show that the dimer covering condition
allows the same extension onto open neighborhoods in the full projector space. This
proves part 1 of theorem 2.
3
k −1
The above analysis is essentially local, showing that the set W ⊂ (CP2
)M
of PRODSAT projector choices is nonempty and full dimension; to extend product
satisfiability to the entire projector manifold, we need to appeal to some basic results
in complex projective geometry: 1. Any Zariski-closed full dimension subspace of an
irreducible complex projective space (such as (CP2
k −1
)M ) is in fact the whole space
and 2. W is Zariski-closed. This shows part 2 of theorem 2. For more details, see
Appendix 3.A.
3
For those who prefer a more rigorous looking style, we note that the first order perturbative
calculation performed in this section is identical to checking the conditions under which the implicit
function theorem provides a smooth map from projector space to the manifold of satisfying product
states near a known solution.
53
Finally, we note that the converse statement also holds (also shown in Appendix 3.A):
Theorem 3. If G does not have dimer coverings, (G, Πm ) is not PRODSAT for
almost all Πm .
3.3.3
Counting product states at M = N
We now comment on the marginal case when M = N . In this case, one expects the
satisfying product states, when they exist, to be discrete. Let us return to projectors
of product form and consider a choice φ0 in which no local projectors on a single site
n are parallel, as mentioned above. In this case, it is clear that there is a one-to-one
correspondence between dimer coverings c of G and product state solutions ψc . The
number of product states at the special point φ0 is given by the number of dimer
coverings.
Now, one can pick any of the ψc and repeat the local perturbative analysis around
φ0 . This will show a unique way to continuously extend ψc in an open neighborhood
of φ0 . Thus, at least on open neighborhoods within projector space, the number
of product states is bounded below by the number of dimer coverings. Since the
problematic points in φ space for this local analysis are actually degenerate points
with higher satisfiability, this should be a lower bound on the number of product
states for any φ and it should be the exact counting almost everywhere.
With a handle on all of these product states, one might hope to make some sharp
statements about the QSAT dimension of G – that is, the generic RG = | ker(H)| for
the graph G. Let P S = Span{ψc } be the span of the dimer covering states at some
generic product projector point φ0 . If RP S = |P S| of the ψc are linearly independent,
they will remain so on an open neighborhood of φ0 , since the determinant of their
overlap matrix is a smooth function. We recall that the QSAT dimension RG takes
54
its minimal value almost everywhere on projector space [78]. Since RP S lower bounds
RG on an open neighborhood, it must therefore lower bound RG for all projectors φ.
Using the dimer covering states to characterize the full SAT subspace ker(H) for
a given graph G requires answering two questions: 1. Are the product states linearly
independent? and 2. Do they span ker(H)? For some simple families of graphs,
it is possible to prove the linear independence of the dimer states by exploiting the
geometry of their dimer coverings. On the other hand, it is easy to construct graphs
where the dimer states are not independent: for a fully connected bipartite factor
graph, the number of dimer coverings is N !, which is significantly greater than the
size 2N of the Hilbert space.
The second question is harder to address analytically. Product states do span the
kernel for many graphs that we have studied numerically, but we also know examples
of small graphs for which they do not. Rather, RG can be strictly greater than the
number of dimer coverings. Indeed, H restricted to product projectors may have an
even greater satisfying dimension than it does for general entangled projectors, even
though the number of product states should be no greater.
3.4
Random k-QSAT
There are three flavors of arguments that have been used to pin down the phase diagram of k ≥ 3-QSAT: construction of satisfying product states [77, 78]; combinatorial
upper bounds on the zero state degeneracy [23, 78]; and, a non-constructive invocation of the quantum Lovász local lemma to establish the entangled SAT phase [8].
All three ultimately rely on establishing a correspondence between some geometric
feature of the random graph G (or its subgraphs) and the properties of zero energy
states for generic instances through geometrization.
55
For sufficiently small α, the important interaction graphs have vanishing cores 4 .
In this regime, it is possible to construct the interaction graph by adding in edges one
by one where each additional edge brings with it at least one new leaf node. Thus by
a generalization of the transfer matrix arguments for k = 2, it is possible to construct
a satisfying product state on the full hypergraph; the details are in Sec. 3.4.1.
More generally, we can use the generic dimer covering characterization of Sec. 3.3.2
to extend the PRODSAT regime to larger α than the generalized transfer approach –
→
and it also shows that beyond αps (k) k→∞
1− , there are no satisfying product states.
We will work out this application in Sec. 3.4.1.
The existence of an UNSAT phase follows directly from the existence of a classical UNSAT phase by the geometrization theorem, as we have already discussed in
Sec. 3.3.1. At high clause densities, a simple local bound on the generic dimension of
the zero energy subspace provided the first evidence for the existence of an UNSAT
phase. Recently, Bravyi et al (BMR) [23] have dramatically improved this bound by
finding the ground state degeneracy of larger clusters (“sunflowers” and “nosegay”
graphs) and determining their prevalence within the large random graph. Using these
techniques they have placed an upper bound, αc+ , on the SAT/UNSAT transition of
αc+ ≈ 3.594 for k = 3 and with an exponential scaling αc+ ∼ 2k for large k. These
show that the UNSAT transition for QSAT is at strictly lower clause density than
in the classical ensemble for all k. That is, the quantum ensemble is indeed more
frustrated than the classical one. We will review these arguments in Sec. 3.4.2.
The gap between PRODSAT phase and the best UNSAT bounds is distressingly
large: αps = O(1) versus αc+ = O(2k ). Moreover, the existence of a satisfiable regime
above αps would provide a simple ensemble of large graphs with intrinsically entangled
satisfying states. We note that for k = 2, such a regime does not exist. In Sec. 3.4.3,
4
The core of an interaction graph is the maximal subgraph in which every node participates in at
least two interactions. Alternatively, it is the subgraph remaining after pruning, in which dangling
leaf nodes are removed (along with their attached interactions) iteratively until no leaves are left.
56
we look for the existence of this phase numerically for small k and analytically for
large k, where Ambainis et al [8] have proven the existence of such an entangled SAT
phase for k ≥ 12 using their quantum Lovász local lemma.
Finally, we comment on the satisfaction of the promise gap in Sec. 3.4.4 and on
the generalization of QSAT to higher rank projectors in Sec. 3.4.5.
3.4.1
Existence of PRODSAT
Generalized transfer matrix
We prove the existence of a PRODSAT phase at low clause density for rank 1 kQSAT by constructing product states of zero energy for arbitrary interaction graphs
containing a core with a satisfying product state. Since the core of a random graph
vanishes for α < αhc (k), this proves the existence of a satisfiable phase below the
emergence of a core.
Lemma 4. Suppose that H is an instance of QSAT on N qubits with a satisfying
product state |Ψi. Let H 0 = H + Π be an instance of QSAT on N 0 > N qubits where
Π is a QSAT projector touching at least one qubit not among the original N . Then
H 0 has a satisfying product state |Ψ0 i.
Proof. Assume first that the interaction Π adjoins only one dangling qubit (untouched
by H), say qubit 0. Let the components of |Ψi be Ψi1 ···iN . The key observation here
is that the projector Π with k − 1 of its qubits fixed in a product state of N qubits
uniquely specifies the state of the dangling qubit 0. If φi0 ···ik−1 is the vector onto
which Π projects, then the state of qubit 0 must be perpendicular to φ∗i0 ···ik−1 Ψi1 ···ik−1 .
i0
0
Thus, contraction with i0 i0 defines a k−1 to 1 transfer matrix Ti10···ik−1 generalizing
the Bravyi transfer matrix of 2-QSAT. The new product state given by |Ψ0 i = |ψ 0 i ⊗
|Ψi where ψi00 = Tii10···ik−1 Ψi1 ···ik−1 satisfies Π. Moreover, since H does not act on qubit
57
0, H |Ψ0 i = |ψ 0 i ⊗ H |Ψi = 0. Thus H 0 = H + Π is satisfied by the product state |Ψ0 i
on N + 1 qubits.
Finally, if Π adjoins more than one dangling qubit, simply fix all but one of these
to an arbitrary state and then apply the above transfer matrix procedure.
Theorem 5. Suppose that H is an instance of random QSAT and let HC be the
restriction to its core. If there is a satisfying product state on HC , then there is
almost surely a satisfying product state on H.
Proof. Let Πi , i = 1 · · · M (G) − M (C) be the sequence of projectors removed in the
process of stripping leaves from the graph G of H to produce the core. At each stage
of this process, the removed projector Πi has at least one dangling qubit. Thus, if
we iteratively reconstruct H from HC by adding back the Πi in reverse order, we can
apply the lemma at each stage to lift the product state on HC to a product state on
H.
Dimer coverings of random graphs
From Sec. 3.3.2, we know that the existence of dimer coverings on G implies quantum satisfiability – and that their nonexistence shows satisfying product states are no
longer generic. For the random graph ensemble, we can therefore identify a PRODSAT threshold αps with the threshold for the presence of dimer coverings αdc . This
turns out to be precisely the point at which the core of the random graph reaches the
critical density Mhc = Nhc , beyond which it is certainly impossible to dimer cover the
clauses. In general, the core emerges at αhc < 1 with a clause density less than 1 and
as α grows, it gets denser. Only at αdc > αhc does it become critically constrained.
See Table 3.1 for numeric values of αdc .
58
In order to show that dimer coverings exist below αdc , we rely on a somewhat
indirect argument based on the results of Mézard et al. [92] regarding the random
k-XORSAT problem, or, in physics parlance, the p-spin Ising glass (with p = k). In
this problem, N classical bits live on an interaction graph G whose (hyper-)edges
define parity check constraints. That is, for each edge m, we must choose Jm ∈ {0, 1}
and then determine whether the following linear constraint equations are satisfiable:
xm1 + xm2 + · · · + xmk = Jm
(3.18)
where xn are the bits and all arithmetic takes place in the field Z2 .
This defines a linear system of inhomogeneous equations of exactly the same form
as what we discovered when linearizing PRODSAT:
Amn xn = Jm
(3.19)
where A is the node-edge adjacency matrix of G. In this case, A is surjective if
and only if all choices of Jm result in satisfiable instances of XORSAT. Moreover, if
A is surjective over Z2 , so is the sparse matrix Ωmn = φmn Amn over C with arbitrary components φmn : if Ω were not generically surjective, the rank discriminating
polynomials discτ (Ω) of Eq. (3.17) would be identically zero, including when mapped
homomorphically to Z2 .
Mézard et al show that the random XORSAT problem is unfrustrated for α < αdc .
Indeed, they show precisely that with probability going to 1 in the thermodynamic
limit, the random graph G is XORSAT for all choices of Jm . In particular, this implies
that A is surjective for α < αdc and therefore that Ω is as well, dimer coverings exist
and the QSAT problem is satisfiable. Meanwhile, for α > αdc , M > N on the core
and the dimer coverings are gone.
59
k αhc
2 0.5
3 0.82
4 0.77
5 0.70
∞ 0+
Quantum
Classical
−
+
+
αps
αc
αcl
αcl
0.5
0.5
1
1
0.92
3.59
3.52
4.49
0.98
7.98
7.91
10.23
0.99
16.00
18.79 21.33
1− 2k−1 ln 2 2k ln 2 2k ln 2
Table 3.1: Summary of critical values for various random k-(Q)SAT properties. The
values for the emergence of a core αhc and for the dimer coverability/product satisfiability αps = αdc of the random interaction graph may be found in [92]. αc+ (3) comes
from the BMR ‘nosegay’ bound [23] while the values at larger k and in the large k
limit come from the ‘sunflower’ bound. For comparison, we include the best known
rigorous bounds on the classical satisfiability transition [1, 34].
We note that this significantly improves the PRODSAT lower bound given by the
emergence of the generalized transfer matrix approach of the previous section, αhc .
As k becomes large, αhc approaches zero while αdc approaches 1 (from below).
3.4.2
Existence of UNSAT
First moment bound
For a given random graph, let us construct H via the sequence Hm =
m
P
j=1
I
Πφjj , i.e. we
add the projectors one at a time in some order. Clearly H = HM . Let Rm be the
dimension of the satisfying subspace for Hm ; evidently R0 = 2N . At each step, if the
next added projector involves a set of qubits that were not previously acted upon,
then Rm+1 = Rm (1 − 21k ) as we may simply implement the projection by reducing the
size of a basis that can be factorized between the target qubits and all others. It is
intuitively plausible that this is the best we can do—in cases where the target qubits
are entangled with the rest of the system we should expect to lose even more states,
i.e.
Rm+1 ≤ Rm (1 −
60
1
)
2k
(3.20)
in general. A proof that this bound holds with probability 1 for projectors randomly
chosen according to the uniform Haar measure is contained in [78]. From this we
conclude that
RαN ≤ 2N (1 −
Thus, for α > −1/ log2 (1 −
1
)
2k
1 αN
) .
2k
(3.21)
∼ 2k the problem is asymptotically almost always
UNSAT.
We note that this bound is actually weaker than the one we have already discovered
using geometrization in Sec. 3.3.1. It goes by the name of the ‘first moment bound’
because it corresponds exactly to a ‘first moment bound’ that was first derived for
classical random k-SAT by taking the expectation (first moment) of the number of
satisfying states [50].
Sunflower and nosegay bounds
Bravyi et al [23] generalized the reasoning underlying the first moment bound to
derive a much stronger bound on RG for random graphs G. The essential idea is to
build the graph G up out of small subgraphs where each additional subgraph cuts
down the satisfying space by more than what each individual projector does. This
relies on a generalization of Eq. (3.20):
R(G ∪ H) ≤ 2−t R(G)R(H)
(3.22)
where G and H are interaction graphs on N and t ≤ N qubits respectively and
G ∪ H represents the graph G with a copy of H pasted on t of its qubits. We recover
Eq. (3.20) by taking H to be a single k-qubit projector.
Bravyi et al calculate R(H) exactly for subgraphs H which are small clusters of
overlapping projectors – the aforementioned sunflowers and nosegays are simply the
flowery graphs which they consider. The UNSAT bounds at various k follow by then
61
analyzing the expected number of sunflowers and/or nosegays that a random graph
at density α ought to contain. We refer the interested reader to their paper [23] for
more details of this elegant construction.
Although the flower bounds are sure to be overestimates of the true UNSAT
transition, they are quantitatively important because they lie below the best lower
bounds for the classical k-SAT transitions. This confirms the expectation that the
quantum ensemble is strictly more frustrated than the classical one.
3.4.3
Existence of SAT-UNPRODSAT
In the previous sections, we argued that random k-QSAT is PRODSAT for all k,
at clause densities α < αdc (k) = αps (k) with αps (2) =
1
,
2
αps (3) = .92 . . . and
αps (k → ∞) → 1− . What happens for α > αps (k)? For k = 2, as the ensemble goes
UNPRODSAT, it immediately goes UNSAT as well. For k ≥ 3, no such result exists.
Rather, the best upper bounds on the extent of the SAT phase, due to BMR [23], are
well separated from the PRODSAT transition: a detailed analysis of the equations
they derive yields the upper bounds of Table 3.1. The large-k asymptote of this bound
is αc+ (k) = 2k−1 ln 2 + O ln1k , which is exponential in k, leaving plenty of space above
αps = 1− for a SAT–UNPRODSAT phase.
For k ≥ 13, Ambainis et al [8] have answered this question in the affirmative: the
SAT regime extends to
αc > 2k /(12 · e · k 2 )
which is significantly beyond the PRODSAT transition at αps ≈ 1. Their result follows
from their quantum generalization of the so-called Lovász local lemma of probability
theory and is non-constructive in the sense that it provides no means of constructing
the satisfying states. This is perhaps not surprising since these states are necessarily
62
entangled and need not have a succinct classical construction. We will review the
idea of the Lovász construction in Sec. 3.4.3.
This still leaves open the question of a SAT–UNPRODSAT phase at smaller k.
In Sec. 3.4.3, we present some numerical pointers regarding the existence of such a
phase at k = 3, 4.
Quantum Lovász local lemma
The most recent technique that has been used to fill in the phase diagram of Fig. 3.1
is the development of a quantum version of the Lovász local lemma by Ambainis et al
[8]. This lemma provides a nonconstructive proof that satisfying states must exist for
k-QSAT instances built out of interaction graphs with sufficiently low connectivity –
that is, graphs in which the degree of every qubit is bounded by 2k /(ek). The QSAT
ensemble has average degree αk but the degree distribution has unbounded tails.
By cutting the graph into low and high connectivity subgraphs, using the product
state characterization on the high connectivity part and the Lovász lemma on the
low connectivity part, and carefully glueing these results back together, Ambainis et
al indeed showed that satisfying states exist for α < 2k /(12ek 2 ).
For sufficiently large k, this result proves that αs ≥ O(2k /k 2 ) αps , establishing
the entangled SAT regime indicated in Fig. 3.1.
We now sketch the idea behind the classical and quantum Lovász local lemmas.
Suppose we have some classical probability space and a collection of M events
Bm , each with a probability of occurring P (Bm ) ≤ p < 1. We think of these as
low probability ‘bad’ events, such as “the m’th clause of a k-SAT instance is not
satisfied by σ” given a uniformly chosen configuration σ. In this particular case,
P (Bm ) = p = 1/2k for all clauses m. If there is a positive probability that no bad
event comes to pass, then there is clearly an overall assignment of σ which satisfies
all of the clauses. Thus, we would like to show this probability is positive.
63
If the events Bm are independent, this is clearly possible:
P r(
^
m
¬Bm ) =
M
Y
(1 − P r(Bm )) ≥ (1 − p)M > 0
(3.23)
m=1
In the k-SAT example, clauses are independent if they do not share any bits – thus
this argument provides us the rather obvious result that k-SAT instances composed
of only completely disconnected clauses are satisfiable. On the other hand, if the
events Bm are dependent, it is clear that we can make
^
P r( ¬Bm ) = 0.
(3.24)
m
For instance, simply take a 3-SAT instance with 3 qubits and 8 clauses, each of
which penalizes a different configuration. These clauses still have individually low
probability (p = 1/2k ) but at least one of them is violated by any configuration.
The classical Lovász local lemma [40] provides an elementary method for relaxing
the independence requirement a little bit. In particular, if each event Bm depends on
no more than d other events Bm0 where
ped ≤ 1
(3.25)
(e ≈ 2.7182 . . . ) then the local lemma tells us that there is indeed a positive probability that no bad event happens:
^
P r( ¬Bm ) > 0
(3.26)
m
This means that for connected k-SAT instances of sufficiently low degree, Lovász
proves the existence of satisfying configurations.
64
In the quantum generalization of the Lovász lemma, probability is replaced by
the relative dimension of satisfying subspaces. That is, for a QSAT projector Π of
rank 1, the “probability of the clause being satisfied” is
Dim(SAT )
Dim(H)
=
2k −1
2k
= 1−
1
.
2k
With the right definitions in hand, the generalization is also elementary and the result
looks nearly identical to the classical case. However, now a positive probability that
all projectors are satisfied tells us that there exists a (potentially quite entangled)
quantum state of an N -qubit Hilbert space which satisfies the instance of k-QSAT.
The Lovász local lemma is nonconstructive because it works by bounding (from
below) the satisfying space degeneracy as the graph G is built up, so long as each
additional clause does not overlap too many other clauses. In some sense this is dual
to the arguments used to prove the UNSAT phase exists by bounding this degeneracy
from above, but the technical details are somewhat more subtle since they require
a more careful consideration of the interaction between additional clauses and the
existing constraints.
In the last few years, computer scientists have developed a constructive version of
the classical Lovász local lemma. That is, there are now proofs that certain probabilistic algorithms will actually efficiently construct the Lovász satisfying states [99].
Recent work suggests that a quantum generalization of this constructive approach
may also be possible [9].
Numerical search for SAT–unPRODSAT instances
First, we have directly searched for instances – irrespective of their probability in the
random graph ensemble – which are SAT–unPRODSAT. Our search for satisfying
assignments proceeded via (i) exact diagonalization of systems with up to 14 qubits
and (ii) a numerical nonlinear equation solver for finding product solutions. Our
results were all consistent with the picture of product states outlined in previous
sections. In addition, we found graphs with satisfying assignments with M > N .
65
a) k=2
b) k=3
32
PRODSAT
2
b+1
2
SAT,
unPRODSAT
b
unSAT
100
Rank of RDM
24
20
16
unSAT
10
12
8
Tree
4
Loop
0
1
2
3
4
5
b qubits in block
Rank Histogram
28
1
0
4
8
12 16 20 24 28 32
Rank of 5 qubit RDM
Figure 3.4: Rank analysis of reduced density matrices (RDM) in a ground state: (a)
k = 2: Rank of the reduced density matrix containing b = 1, . . . , 5 qubits for different
types of interaction graphs. (b) k = 3: Histogram of ranks of b = 5 reduced density
matrices for all 252 different partitions (N = M = 10) for three different instances.
In these cases method (ii) failed to turn up any product ground states. We also
diagnosed directly the presence of non-product ground states by considering the rank
of the reduced density matrix corresponding to a ground state after an arbitrary
portion of the system is traced over, cf. Fig. 3.4. The resulting rank of the density
matrix is bounded above by the number of zero-energy states, provided these are
spanned by product states (c.f. Appendix 3.B for a proof). As an illustration, we
show the corresponding rank for the 2-SAT problem in Fig. 3.4(a) for reduced density
matrices of up to five qubits from instances containing 13 qubits. While the SAT
instances show the expected rank for product states, the unSAT instance has full
rank, suggesting generically entangled wavefunctions in the k = 2 unSAT phase. In
Fig. 3.4(b) we performed a similar analysis for k = 3. We determined rank histograms
for all 252 five-qubit reduced density matrices of N = M = 10 systems for three
66
different instances
5
. The PRODSAT instance leads to a histogram where the rank
is bounded by the number of satisfying ground states (16 for this instance). The
SAT–unPRODSAT instance shows ranks which are both smaller and larger than
the number of satisfying ground states (23 for this instance), but do not reach the
maximum of 25 = 32. Finally, the unSAT instance has full rank 32 on the two-core
(the rank 8 occurring for some partitions is due to a satisfied dangling clause). We
do indeed find that, in the presence of M > N on a (sub)graph, entangled states
necessarily appear in any basis of the ground states.
Finite size scaling
Where is the the SAT/unSAT transition αc ? Numerical studies on finite-site systems
offer limited information as the properties of small random graphs are quite different
from those of large ones. We consistently find the presence of SAT instances with
N > M : for 3-SAT, we investigated 101 random graphs for sizes with up to 13 sites
using complete diagonalization and up to 20 sites using an iterative diagonalization
technique, see left panel of Fig. 3.5. While complete diagonalization is limited by the
growing matrix size but otherwise numerically stable, the iterative Lanczos procedure faces challenges when resolving small energy gaps occurring at the SAT-unSAT
transition. This leads to some instances becoming undecidable, i.e. not converging
within several thousand iterations. The presence of undecidable instances is denoted
by filled symbols 6 . For 3-SAT, we find that curves for larger N nicely converge,
suggesting αc ≈ 1 ± 0.06, where the error bar is estimated from the largest N in
Fig. 3.6. For 4-SAT, instances are SAT for larger values of ∆ = M − N (for fixed
5
The three N =M =10 graph instances are: a) PRODSAT: {{3,5,8}, {5,6,7}, {0,6,8}, {1,3,5},
{0,2,5}, {1,3,9}, {1,4,9}, {0,1,5}, {2,3,7}, {0,1,2}} b) SAT-unPRODSAT: {{0,1,5}, {0,4,5}, {2,5,8},
{5,7,9}, {5,6,7}, {5,7,8}, {6,8,9}, {0,3,5}, {4,6,8}, {2,4,9}} c) unSAT: {{2,7,8}, {0,4,7}, {4,5,6},
{1,5,6}, {1,6,9}, {3,5,7}, {0,3,7}, {5,6,7}, {1,3,7}, {1,6,7}}
6
Among the 101 graphs we found the following maximum number of undecidable instances:
N = 14, k = 3, max 4 undecided; N = 16, k = 3, max 9 undecidable; N = 14, k = 4, max 24
undecidable; N = 16, k = 4, max 57 undecidable; N = 18, k = 4, max 62 undecidable; N = 20, k = 4,
max 58 undecidable.
67
k=3
k=4
1
0.6
0.4
0.2
0
0
0.8
N=6
N=8
N=10
N=11
N=12
N=13
N=14
N=16
N=18
N=20
0.5
0.6
0.4
0 2 4 6 8
∆=M-N
1
1.5
2
0
M/N
0.5
SAT Ratio
SAT Ratio
0.8
1
0.2
1
1.5
0
2
M/N
Figure 3.5: Probability to find SAT instances among 101 random graphs for (a) k = 3
and (b) k = 4. Numerical results obtained by complete diagonalization for N ≤ 13
and iterative diagonalization for N = 14, 16 (k = 3, 4) and 18, 20 (k = 4). For the
latter system sizes filled symbols indicate the occurrence of (numerically) undecidable
instances.
N ), even though ∆ appears to decrease with N (inset in right panel of Fig. 3.5). The
finite size scaling shown in Fig. 3.6 is also compatible with αc ≈ 1, but with a larger
uncertainty of ≈ 0.2.
Thus, we find many examples of generically SAT–unPRODSAT graphs G and
even some evidence that the unSAT phase transition in the random ensemble lies at
α > αps .
3.4.4
Satisfying the promise
As in the case of 2-QSAT, physical experience suggests that the ground state energy
for k-QSAT should be extensive with small fluctuations above the satisfiability transition. If this is true, the promise is satisfied with probability exponentially close to 1,
as in the k = 2 case. Unfortunately, we do not know a set of unsatisfiable subgraphs
analogous to the figure eight’s that might be used to provide a more rigorous bound.
68
2
1.8
k=3
k=4
1.6
αc(N)
1.4
- -- - - - - -- - - -- - - -
1.2
1
0.8
- - - - -
-
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
k
2 /N
Figure 3.6: Finite size scaling of αc (N ) for k = 3, 4 obtained from the data displayed
in Fig. 3.5. The empty symbols denote α values where the SAT ratio is 0.5. The
lower (upper) ticks correspond to the largest α values with SAT ratio > 0.9 (smallest
α values with SAT ratio < 0.1).
3.4.5
QSAT at rank r > 2
We now briefly consider the extension of k-QSAT to (k, r)-QSAT, in which the projectors ΠI have rank r, i.e. they penalize a uniformly chosen r-dimensional subspace of
the 2k dimensional k qubit Hilbert space. The main results regarding k-QSAT generalize naturally: satisfiability is almost surely only dependent on the underlying graph
αN
(r)
and the weak bound on the ground state degeneracy becomes RαN ≤ 2N 1 − 2rk
,
(r)
implying a bound αc ≤ −1/ log2 1 − 2rk . However, there need not be a SAT phase
at all: if r = 2k , the projectors ΠI are each the identity and the ground state energy
is αN > 0 for any positive α. More generally, there is some critical rank rc above
which the SAT phase disappears.
We bound rc above by 2k /2 by exhibiting an unsatisfiable subgraph of the random
hypergraph that arises asymptotically almost surely (N → ∞) in the random graph
ensemble even at small α. Consider a chain with two clauses and one shared qubit.
69
Classically, this corresponds to a two-clause problem on 2k − 1 bits where we allow
each clause to forbid r = 2k /2 configurations. Now let the first clause forbid all
configurations in which the shared bit is 0 and the second clause all configurations
in which it is 1. This classical problem is clearly unsatisfiable and therefore so is
any quantum problem on this subgraph w.p. 1. Indeed, since there are extensively
many such small chain components in the hypergraph, each with an independently
chosen O(N 0 ) ground state energy, this provides an extensive lower bound on the
total ground state energy.
The bound rc ≤ 2k /2 is not tight however. For example, for k = 3, an open chain
of length four with rank 3 < 23 /2 = 4 is quantum mechanically unsatisfiable w.p. 1,
as can be checked numerically, and therefore so is the ensemble for (k, r) = (3, 3).
However, there is no classically unsatisfiable problem on this chain and it is harder
to construct a rigorous bound that scales with k using this starting point.
Finally, we note that the problem that of higher rank QSAT on chains (rather
than random graphs) has been considered in some detail by Movassagh et al [101].
The chain geometry provides a natural framework within which to look for weakly
entangled matrix product states in the generic QSAT problem.
3.5
Open questions
Many questions regarding quantum satisfiability remain open and it is to these that
we turn in closing:
1. Does the SAT–UNPRODSAT phase exist for all k ≥ 3 or does it really only
arise at some larger k?
2. Is there a thermodynamic phase transition (i.e.. a non-analyticity in the zero
energy entropy) at the entanglement transition? Are there other analogs of
the clustering transitions? Indeed, is there a meaningful sense in which the
70
ground state space is clustered – composed of macroscopically distinct clusters of
states? This is reminiscent of the many-body localization phenomena argued to
exist in classical hard optimization problems subject to the quantum adiabatic
algorithm [7].
3. Does an easily checked classical graph property encode generic quantum satisfiability? Does a straightforward function compute the generic dimension RG ?
4. What can be said about higher rank projectors? The product satisfiability
condition translates straightforwardly to the higher rank case by simply treating
a rank r projector as a stack of rank 1 projectors. Thus, higher rank QSAT
instances may naturally yield more entangled ground states than the rank 1
case. Indeed, preliminary work indicates that this is the case [101].
5. What happens to the energy density above the UNSAT transition? What is
the structure of the finite temperature phase diagram? Does QSAT anneal in
finite time to any given temperature? Does it have a localization transition? Is
QSAT hard to approximate?
3.A
Formal PRODSAT proof
We expand on the complex algebraic details of theorems 2 and 3. That is, we show
that interaction graphs G with dimer coverings are PRODSAT for any choice of projectors {Πm }M
m=1 and conversely that G without dimer coverings are not PRODSAT
for almost all such choices. The relevant algebraic geometry background can be found
in [94, 102].
k −1
Proof. Let P = (CP2
)M be the space of k-qubit projectors, S = (CP1 )N be the
N
space of product states, and V ⊂ P × S be the space of pairs of ({Πm }M
m=1 , {|ψn i}n=1 )
such that |Ψi = |ψ1 i ⊗ · · · ⊗ |ψN i satisfies the projectors Πm . We note that P , S
71
and P × S are each irreducible complex projective spaces in the Zariski topology
and that V , as the zero-locus of collection of homogeneous algebraic constraints, is a
closed subspace of P × S. Let π : P × S → P be the projection map onto the first
component. Since S is projective, this map is closed.
We are interested in the set of projectors W ⊆ P which are satisfiable by product
states. These are precisely the projectors in the image of V under the projection π:
W = π(V )
(3.27)
Since π is closed, W is closed. By the local arguments of Sec. 3.3.2, we know that
W is full dimension and nonempty. Since P is irreducible and projective, it has no
proper full dimension closed subspaces; i.e.. W = P .
Conversely, if G does not have dimer coverings, there exist choices of product
projectors Πm which are clearly unsatisfiable by product states (see Eq. (3.14)). Hence
W is a closed proper subspace of P and cannot be full dimension. Thus, W has
codimension at least 1 in P and almost all choices of projector Πm will fail to be
product satisfiability.
3.B
Reduced density matrix diagnostic for product states
Let us consider a graph Λ and a wavefunction |Ψi, which is expressible as a sum of
wavefunctions each of which is a product wavefunction on the sites i ∈ Λ. Let γ ⊂ Λ,
(γ)
and let ρ = |ΨihΨ| and ρ̃γ = TrΛ\γ ρ. Then rank ρ̃γ ≤ N0
(γ)
number of linearly independent product states, and N0
when restricted to γ.
72
≤ N0 where N0 is the
the dimension of their span
Let us label a product basis for the ground state manifold by: {Φα |α = 1 . . . N0 }
s
with |Φα i = ⊗ni=1
|φαi i, hφαi |φαi i = 1. Here ||Λ|| = ns . Thus, hΦα |Φα i = 1 (normalized)
Qn s D α β E
and hΦα |Φβ i = i=1 φi |φi (not necessarily orthogonal).
A general ground state is
N
0
1X
|Ψi =
λα |Φα i
ℵ α=1
with
2
ℵ =
for which
N0
X
α,β=1
λα λ∗β
ns
Y
i=1
φiβ |φiα
(3.28)
,
(3.29)
N0
1 X
ρ = |ΨihΨ| = 2
λ∗ λα |Φα ihΦβ | .
ℵ α,β=1 β
(3.30)
Tracing over the qubits on subset Λ \ γ of the set Λ yields:
N0
h
i Y D
E
1 X
β
β
∗
α
α
ρ̃γ = 2
λ λα [⊗i0 ∈γ |φi0 i] ⊗j 0 ∈γ hφj 0 |
φk0 |φk0
ℵ α,β=1 β
0
.
(3.31)
k ∈Λ\γ
ρ̃γ is at most an N0 × N0 matrix, with at most rank N0 irrespective of the choice of
γ.
(γ)
If dim (span {⊗i0 ∈γ |φαi0 i|α = 1 . . . N0 }) = N0 < N0 , then we can write, for α = N0
PN0 −1 (N0 )
0
(say): ⊗i0 ∈γ |φN
[⊗i0 ∈γ |φαi0 i], so that terms like
i0 i =
α=1 µα
0 −1
h
i
NX
β
N0
N0
(N0 ) ∗(N0 )
α
[⊗i0 γ |φi0 i] ⊗j 0 ∈γ hφj 0 |
⊗i0 γ |φi0 i ⊗j 0 γ hφj 0 | =
µα µβ
,
(3.32)
α,β=1
take on the same form as the ones in (3.31). Using such substitutions repeatedly,
(γ)
one finds that the reduced dimensionality manifestly has a rank of at most N0 .
(γ)
In particular, for the case of 2-QSAT and Λ being a tree (||Λ|| + 1 = N0 ), N0
=
||γ|| + 1. When a qubit being traced out is a dangling one, this result is obvious. In
the case of an internal qubit (labelled 2, say), we note that the Bravyi construction
73
(γ)
on Λ induces one on γ via >13 = >12 >23 , where >ij represents the Bravyi transfer
matrix between for the projector ij, so that the resulting state counting is that of the
smaller graph.
74
Part II
Cavity method and the Bethe
lattice
75
The appearance of finite connectivity trees (e.g. Cayley trees, Bethe lattices) in
the study of spin glasses has a long history that goes back to the first papers on
the Sherrington-Kirkpatrick model [111, 115]. Thouless, Anderson and Palmer [115]
showed by means of a diagrammatic expansion of the partition function that in the
spin-glass phase the mean-field theory is defined by the diagrams which describe an
infinite tree with connectivity q 1. They then wrote the mean field equations for
such a tree and simplified the results in the small coupling, large connectivity regime
√
(since in the SK model q = N and the couplings Jij ∝ 1/ N ). Those mean field
equations are known as the TAP equations and the peculiar characteristics of their
solutions, in particular their large number [25], were an important indicator of the
complexity of the spin glass phase that arose in parallel with remarkable developments
in the study of the replicated free energy [33, 104, 106, 105, 107].
Many authors have since studied the problem of a spin glass on the Bethe lattice
[117, 19, 75, 54, 55, 29, 27] with two distinct motivations. The first of these has been
to attempt to find a model of a short ranged spin glass where one can rigorously assess
Parisi’s picture for the organization of the Gibbs states in short-range systems [47, 46].
The second has come from computer science whereby a set of optimization problems
can be recast as frustrated problems on random graphs with the local connectivity of
a tree [97, 91, 93].
Despite much work, the Bethe lattice has not yielded a decisive verdict for or
against the Parisi picture of a multitude of Gibbs states in the ordered phase of a
spin glass although the case for it is perhaps stronger here than on regular lattices
with short ranged interactions. This is tied up with the question of defining the
infinite Bethe lattice limit starting from finite graphs. It is possible to do so either
via a sequence of Cayley trees with random, frustrating, boundary conditions or
via a sequence of random graphs with fixed connectivity. The latter sequence has
frustrating loops of typical size diverging O(log(N )/ log(q − 1)) for a graph with N
76
points and connectivity q and thus locally looks like a tree [17]. The Cayley tree
sequence can always be analyzed in terms of a recursion relation that we review
below and does not appear to lead to the Parisi structure [29, 27]. However, it has
been argued that the random graph problem is fundamentally different and does lead
to replica symmetry breaking [89, 88].
Happily, the perspective provided by spin glass theory on optimization problems
has been quite fruitful regardless. Starting in the early 80’s [67, 51], it became
clear that much was gained by the recognition that various optimization problems
in computer science were equivalent to finding ground states of certain statistical
mechanics problems. A typical example of this connection is given by the (classical)
k-SAT problem, which we have already introduced in Chapter 2. To recap, in the
language of statistical mechanics, bits naturally become Ising spins, the interactions
J become a particular instance of some spin glass energy function and the SAT
question requires determining the ground state energy. The large N limit is a problem
in statistical mechanics and typical-case analysis for k-SAT becomes the disorder
averaged analysis of spin glass theory. In this limit, k-SAT develops several phase
transitions as a function of α = M/N , the number of clauses per bit [98]. With
increasing α, the most salient features are that the problem goes from being easily
solved and satisfiable, to an intermediate glassy phase with many local ground states,
to a typically unsatisfiable phase where the ground state energy density is positive.
A key role in these developments has been played by the so called cavity method —a
complex of analytical and numerical techniques refined recently [89, 88, 97, 91, 93]—
for studying classical spin glasses on tree-like graphs. Applied to the k-SAT problem,
the cavity method suggests the above phase diagram and provides numerical values
for its critical points. Moreover, the technique can be applied to a particular instance
of k-SAT and the information so obtained about the free energy landscape now guides
the search procedure in state-of-the-art k-SAT algorithms [97, 91, 93].
77
In Part II of this thesis, we turn to quantum spin glasses on Bethe lattices, specifically to the problem of extending the cavity method to their analysis. As in the
classical case, there are two distinct reasons to be interested in these systems. There
is the intrinsic interest of the interplay between quantum mechanics and spin glass
behavior, about which the difficulties of the classical case serve as both caution and
enticement. A second motivation now arises from the rapid recent developments in
quantum computing. In particular, the discovery of the ground state of a classical
spin system HJ derived from a computational problem is ideally suited to solution
by the adiabatic algorithm [43], which allows a quantum computer to solve such a
problem by means of an adiabatic change of the parameters in the Hamiltonian. The
adiabatic algorithm would trace a path in operator space starting from a simpler
Hamiltonian H0 and ending at HJ . For instance, H0 could be the Hamiltonian for N
independent spins in a large transverse magnetic field Bt and the path could be to
slowly lower the field and raise the Ising couplings J. Starting from the easily found
ground state of H0 a sufficiently slow evolution will carry the system to the ground
state of HJ . The power of this algorithm is therefore measured in the scaling of the
evolution time with the number of variables N .
This discussion certainly suggests that a careful study of the phase transitions
encountered on the path between H0 and HJ is in order. However, the necessity of
understanding the “deep” quantum spin glass phase far from the phase transition is
also clear. The structure of this phase, especially its nontrivial energy landscape, is
probably as important as the nature of the phase transition. In the following chapters
we take a step toward understanding the quantum spin glass phase by generalizing the
cavity method used to study the classical problem. Compared to the replica method or
direct study of the quantum TAP equations, we believe the cavity approaches outlined
in the next few chapters provide much more physically transparent information. It
78
also may be applied to many other quantum phase transitions on the Bethe lattice,
whether induced by disorder or more typical symmetry breaking.
79
Chapter 4
Introduction to the cavity method
The cavity method is a cluster of techniques and heuristics for solving statistical
models on sparse, tree-like interaction graphs G. In this approach, one determines
the behavior of the model on G by first analyzing the restriction of the model to
so-called cavity graphs. A cavity graph G\{i} is formed by carving site i out of G:
The neighbors of i, ∂i, which now sit at the boundary of the cavity in G\{i}, are
called cavity spins. The central assumption of the cavity method is that cavity spins
are statistically independent in the absence of site i because they sit in disconnected
components of G\{i}. This assumptions massively simplifies the evaluation of observables in such models, ultimately leading to efficient procedures for finding ground
states, evaluating correlation functions and determining thermodynamic free energies,
phase diagrams, and clustering phenomena in models with quenched disorder.
In the following sections we will introduce the most salient features of classical
cavity analysis in the context of every physicist’s favorite statistical model, the Ising
80
magnet. The Ising ferromagnet on an infinite tree makes the Bethe-Peierls mean field
theory of ferromagnetism exact. The cavity approach reproduces this result; more
generally, the local cavity analysis that we sketch below is exact on trees, even in the
presence of quenched disorder. In this context, the self-consistency equation of the
Bethe-Peierls ferromagnet lifts to a distributional fixed point equation, known as the
replica symmetric (RS) cavity equation.
Replicas and replica symmetry breaking play an important role in an alternative,
somewhat more formal, approach to studying statistical models with quenched disorder. In the context of the cavity method, they are merely imported jargon, flagging
the equivalence to the replica approach in various regimes [89, 88, 54, 55]. In the
cavity approach, however, replica symmetry breaking corresponds to the appearance
of multiple (usually exponentially many) solutions of the so-called belief propagation
(BP) equations on models defined on graphs that look locally tree-like but in fact
have long loops. By long loops, we generally mean loops whose length diverges with
the thermodynamic limit – making such models look locally identical to the infinite
limit of simple trees! We will consider the subtleties of the thermodynamic limit for
tree-like graphs in more detail in Sec. 4.3 and Chapter 8. The belief propagation
equations, and the important algorithm which shares their name, will be discussed in
Sec. 4.4.
One of the goals of this thesis is the development of techniques to better understand spin glasses and hard optimization problems in the context of quantum
mechanics. Thus, section 4.5 discusses the generalization of cavity techniques to
quantum models through quantum-classical mappings. This technique is central to
Chapters 5 and 6.
We note that one of the largest difficulties in discussing the cavity method is
notational: cavity equations tend to develop complicated indices and labels despite
81
all efforts to conserve ink. We hope that the discussion below, which uses graphical
representations where possible, reduces this typographic barrier to entry.
4.1
The Bethe-Peierls method for the Ising ferromagnet
We first recall the cavity method in its simplest form: we apply it to a ferromagnetic
Ising model living on a Bethe lattice with connectivity q. There is considerable
subtlety in defining a sensible infinite tree model, but for this simple case, a sequence
of connectivity q Cayley trees with uniform small boundary fields to break the Ising
symmetry will suffice. We will return to alternative definitions of the infinite tree in
Sec. 4.3. In any event, we focus here on the local structure of the model.
The classical Hamiltonian is
H=−
X
Jij σi σj
(4.1)
(ij)
where σi ∈ {±1} are Ising spins, Jij = J > 0 is a uniform ferromagnetic coupling and
the sum is over bonds in the Bethe lattice. To create a cavity, pick a spin σ0 in the
Bethe lattice G and remove it:
(4.2)
Each of σ0 ’s neighbors is a cavity spin, connected to q − 1 spins and sitting at the
root of a branch of the original tree. Notice that in the absence of σ0 these q branches
are entirely independent. Similarly, a cavity spin σ1 mediates the only interaction
between the q − 1 branches sitting above it.
82
σ1
σ2
σ1
Iterate
σ1
σ2
σ0
σ2
σ3
σ1 σ2
σ1
Merge
σ0
σ3
σ2
σ1 σ2
Add Link
Figure 4.1: The three cavity operations. Although we have not labeled them in the
Figure, the new links are J01 , J02 , etc ...
We define three important operations we can perform on graphs with cavity spins:
iteration, merging, and link addition (Fig. 4.1.) For our immediate purposes, the most
important of these is iteration, which takes q − 1 rooted branches of depth l and links
them into a single new branch of depth l + 1 with a new cavity spin at the root.
Thus we can construct an arbitrarily large Cayley tree by iterating inward from its
boundary spins many times followed by a final merge operation to form the center.
Consider the iteration operation: the added spin σ0 receives thermodynamic information regarding each of the q − 1 branches only through the thermal distribution
of the cavity spins σ1 , ..., σq−1 . In the absence of σ0 , each of these Ising variables has
independent statistics characterized fully by its thermal probability distribution:
ψi→0 (σi ) =
eβhi→0 σi
.
2 cosh(βhi→0 )
(4.3)
which defines the cavity field hi→0 . Since σi ∈ {±1}, there are only two possible
configurations of the spin and only 2 − 1 = 1 real numbers are needed to characterize
the probability distribution. In this sense, the cavity field hi→0 is merely a good
parameterization of the distribution ψi→0 (σi ). We emphasize this viewpoint because
it naturally generalizes to models with more complicated local degrees of freedom.
83
We introduce a simple graphical convention for cavity spins:
=
(4.4)
The open circle indicates a spin variable and the wiggly line indicates the effective
field attached to it. With this notation, an iteration operation can be represented:
ψ1
σ1
J01
ψ2
σ2 =
σ0
J02
ψ0
σ0
(4.5)
where the filled circle indicates summing out a spin variable and we have suppressed
explicit reference to the removed cavity site from the subscripts (i.e., ψ1 = ψ1→0 , etc).
More formally, the state of the spin σ0 depends on the state of the q − 1 spins as
1
ψ0 (σ0 ) =
Z
X
exp β
σ1 ,...,σq−1 =±1
where Z is a normalization factor so
equation implies
h0 =
X
i
P
σ0
J0i σ0 σi
!
ψ1 (σ1 )...ψq−1 (σq−1 ),
(4.6)
ψ(σ0 ) = 1. In terms of cavity fields this
q−1
1X
tanh−1 (tanh(βJ0i ) tanh(βhi )) ≡ U ({hi }, {J0i }).
β i=1
(4.7)
Finally, to solve the ferromagnet, we note that all the Jij = J and apply uniform
boundary fields (say slightly positive) to the Cayley tree “leaves”. From this uniform
starting point, we expect to find fixed points for the cavity fields under iteration given
by
h=
q−1
tanh−1 (tanh(βJ) tanh(βh))
β
84
(4.8)
This is precisely the Bethe-Peierls self-consistency equation for a mean field ferromagnetic in a lattice of coordination number q.
4.2
Classical disordered magnet on a Cayley tree
The careful reader will have noticed that until Eq. (4.8), we did not exploit the
uniformity of Jij or hi in the foregoing discussion. With this foundation laid, we can
make short work of the Bethe lattice Ising glass [19, 20]. The Hamiltonian is now
HJ = −
X
Jij σi σj
(4.9)
(ij)
where the Jij are i.i.d. random variables drawn from some distribution P (J). To
make contact with Chapter 5, we consider the ±J spin glass model
1
1
P (Jij ) = δ(Jij − J) + δ(Jij + J),
2
2
(4.10)
but the general analysis works also to study diluted ferromagnets (see Chapter 7),
systems with continuous disorder distributions and many other variations.
The iteration equation (4.6) is still valid for particular realizations of the J, but
since these are random variables, it now defines a Markov process for the cavity fields.
Throughout the graph, these fields will be site dependent random variables, but deep
inside the tree, they ought to be distributed according to a probability distribution
P (h) that represents a fixed point of the Markov process. In particular, the cavity
field distribution will depend only on the depth of a site in the Cayley tree, assuming
i.i.d. boundary fields. This distribution becomes depth independent sufficiently deep
in the tree. The fixed point distribution will satisfy
P (h) =
Z Y
q−1
i=1
dhi P (hi ) hδ(h − U ({hi }, {J0i })iJ .
85
(4.11)
In terms of the spin distribution, this becomes the functional equation
P [ψ] =
Z
q−1
Y
i=1
!
Dψi P [ψi ] hδ[ψ(σ) − ψ0 (σ; {ψi }, {J0i })]iJ
(4.12)
where ψ0 (σ) is given by Eq. (4.6). This is the RS cavity equation.
This distribution is the “order parameter” for the spin glass. It is a δ function at
h = 0 in the high temperature phase. As the temperature is lowered, P (h) broadens to have finite support below some mean field-like phase transition. Defining
1
τ = T /J, this phase transition is located at τc = 1/ tanh−1 ( √q−1
), which may be de-
tected through a linear instability of the paramagnetic solution to broadening at this
temperature. Within the low temperature phase, P (h) develops nontrivial structure
reflecting the random spatial distribution of magnetization in the glass.
The calculation of other observables within the cavity framework is generally
straightforward. For instance, pair-wise connected correlations always decay exponentially: this can most readily be seen by considering the path connecting the pair
in question and recognizing it as a finite temperature spin chain with random fields
coming from each of the attached intermediate branches. This provides a site dependent (random) transfer matrix along the path connecting the two spins which allows
one to calculate correlations at distance l.
The free energy follows from keeping track of the normalizations Zi→j that arise
in the cavity calculations. The resulting free energy per site can be written in terms
86
of P (h) as
Z Y
q
Z 2
q Y
F =
dhi P (hi )Fq+1 −
dhi P (hi )F2 ,
2 i=1
i=1
*
+
q
q
X
Y
X
1
ln
exp(β
J0i σ0 σi )
ψi (σi )
,
Fq+1 = −
β
σ0 ,σ1 ,...,σq
i=1
i=1
J,ψ
+
*
X
1
F2 = −
ln
exp(βJ12 σ1 σ2 )ψ1 (σ1 )ψ2 (σ2 )
.
β
σ ,σ
1
2
(4.13)
(4.14)
(4.15)
J,ψ
This rather complicated looking expression is actually simply the average change in
free energy due to a merge operation minus
q
2
times the average change in free energy
due to a link addition. Graphically,
F =
*


F


ψ1
ψ2
ψ3
 q
− F
 2
ψ1
!+
ψ2
J,ψ
(4.16)
where F of a diagram is the free energy of a system with spin variables given by the
unfilled circles.
It is possible to see that if P satisfies (4.11) then δF/δP (h) = 0. Other expressions
for the free energy F have appeared in the literature but it has been shown that they
are all equivalent to each other, if the consistency equation (4.11) is satisfied. The
role of the free energy in the cavity method is secondary as one does not solve the
variational problem by working on the free energy directly. Rather, one finds the
probability distribution P by analytical or numerical methods and then derives all of
the statistical observables from P .
87
4.3
Cayley trees versus regular random graphs
We have up to this point assumed that the underlying lattice is a tree and that the
removal of a spin to create a cavity completely decouples the neighboring branches.
On such models, the cavity method we have described is exact. An important generalization of the cavity method [88] arises in its heuristic application to locally tree-like
regular random graphs of connectivity q. The ensemble of these graphs is constructed
by assigning uniform probability to all possible graphs of N vertices, such that each
vertex is connected to exactly q links. There are several reasons why statistical models
defined on this ensemble of graphs are interesting:
1. A central property of this ensemble [64] is that typical lattices are locally treelike: a randomly chosen finite neighborhood of the infinite system will be a tree
with high probability. Loops in the graph have a length diverging logarithmically with the size N of the system. This suggests that one can develop an
approximate method to solve statistical models on these graphs based on the
same recurrence equations that are exact on trees.
2. Despite being locally tree-like, regular random graphs do not have any boundary, all sites playing statistically the same role. This is similar to the way in
which periodic boundary conditions impose translation invariance on a finite
Euclidean lattice. Moreover, the free-energy of regular random graph models
is self-averaging with respect to their random character in the thermodynamic
limit. In other words for large enough N a single sample is a good representative
of the ensemble average.
3. Typical graphs are characterized by many large loops of even and odd length;
this strongly frustrates regular (but non-ferromagnetic) ordering patterns. Thus
random graph models produce spin glass phases without need to impose ran88
domly frustrated boundary conditions.1 We we will exploit this observation
especially in Chapter 6.
There is some controversy in the literature as to whether the Bethe lattice should
be thought of as the thermodynamic limit of a sequence of Cayley trees with some
fixed boundary conditions or as the limit of a sequence of regular random graphs.
For models on Euclidean lattices, such a distinction would have little physical consequence: choice of periodic, open or fixed boundary conditions does not effect the
underlying phases of a system, although they might select particular broken symmetry sectors or domain wall structures. In the context of tree-like lattices, however,
the boundary is much more important: the boundary of a tree of any depth is always
a finite fraction of the volume of the tree. Thus it should not be surprising that the
distinction between the different notions of Bethe lattice has a significant impact on
the behavior of statistical models even in the thermodynamic limit. Indeed, this distinction can lead to different physics even for unfrustrated models, as we will explore
in Chapter 8.
We take the view that it is important to know your neighbors and thus we will
always point out which version of a Bethe lattice is under consideration. Except in a
few ferromagnetic models [87], the distinction is important. Despite its lack of loops,
the infinite Cayley tree can still be a useful model for studying frustrated physics by
imposing frustrating boundary conditions – this leads to the replica symmetric (RS)
cavity approach and the RS cavity equations which we derived in Sec. 4.2 above.
Indeed, this is the underlying lattice for the detailed study of the transverse field
Ising glass provided in Chapter 5.
1
See Ref. [88] for a general discussion and Ref. [74] for the explicit computation of the phase
diagram of a classical Ising antiferromagnet.
89
4.4
Relation to belief propagation
The RS cavity equation (4.12) describes the distribution of local observables in thermodynamic models on trees with quenched i.i.d. disorder. Clearly, however, the
cavity iteration relation (4.5) holds identically on any particular realization of the
model, prior to disorder averaging. On finite trees, this observation leads to a linear
time exact method for evaluating the free energy and local statistical observables:
build up the tree iteratively from the boundary inwards, calculating the cavity distributions and associated contributions to the free energy at each step until all nodes
have been included.
For large regular random graphs which are locally tree-like but have long loops,
this approach does not work – there is no boundary from which to work! Nonetheless,
we might hope to heuristically exploit the observation that the cavity relation (4.5)
almost holds when the cavity spins being brought together are almost independent, as
we might expect them to be when they are only connected by long paths. The cavity
iteration relations that arise for a particular finite graph under this assumption are
known as the Belief Propagation (BP) equations and their (efficient) iterative solution
constitutes one of the most important algorithms for approximately solving statistical
models on large graphs.
Let us make these considerations more precise. Consider carving a cavity into a
large regular random graph G with N spins and M edges, as in Fig. 4.2. The statistical
connection between the removed spin σ0 and the rest of the graph is entirely mediated
by the joint cavity distribution
ψG\{0} (σ1 , σ2 , σ3 ) =
1
ZG\{0}
90
Y
X
σj
j ∈{0,1,2,3}
/
e−βHG\{0}
(4.17)
σ2
(σ 2)
ψ 2→0
ψ3→0 (σ3 )
σ 1)
0(
σ3
ψ 1→
σ1
Figure 4.2: Schematic of q = 3 regular random graph with a cavity carved out at spin
σ0 .
That is, the thermal distribution for σ0 in the original model is given by:
ψ0 (σ0 ) =
1 X −β(H01 +H02 +H03 )
e
ψG\{0} (σ1 , σ2 , σ3 )
Z0 σ ,σ ,σ
1
2
(4.18)
3
This generalizes the merge operation of Fig. 4.1 when the cavity spins are not independent. If, on carving out the cavity, the neighboring spins become independent
then the joint cavity distribution factors:
ψG\{0} (σ1 , σ2 , σ3 ) = ψ1→0 (σ1 )ψ2→0 (σ2 )ψ3→0 (σ3 )
(4.19)
Here, the ψi→j is the cavity distribution felt by spin i in the absence of spin j. On
trees, this independence is exact because the cavity spins sit in disconnected clusters;
91
on locally tree-like graphs, the cavity spins are only connected through long (divergent
in system size) paths and thus we might expect Eq. (4.19) to hold approximately.
Thus the objects of interest are the 2M cavity distributions ψi→j , see Fig. 4.3.
These are also known as messages or beliefs: ψi→j (σi ) is a message passed from site
i to site j which indicates site i’s beliefs about what it should do in the absence
of site j, and thus also its beliefs about what site j should do to optimize the free
energy. Crucially, when the cavity distributions are independent, they also satisfy the
iteration relation:
ψi→j (σi ) =
Y X
1
Zi→j
e−βHik (σi ,σk ) ψk→j (σk )
(4.20)
k∈∂i\{j} σk
These are the Belief Propagation (BP) equations and they are really just the graphical
iteration relations Eq. (4.5) in more formal notation.
σ2
ψ 2→0
(σ 0)
(σ 2)
ψ 0→2
ψ
σ1
σ 1)
0(
1→
ψ
σ 0)
ψ0→3 (σ0 )
σ0 ψ3→0 (σ3 )
σ3
1(
0→
Figure 4.3: The belief propagation equations for a graph G involve 2M cavity distributions, or beliefs, ψi→j , one for each of the two directions of a link in the graph.
92
There are linearly many BP equations, each of which is a simple relation involving
a finite summing out procedure on the right to define a cavity distribution on the left.
To solve them, use a standard technique: guess and iterate. Initialize the 2M cavity
distributions randomly and then iteratively sweep back and forth across the graph G,
applying the BP equations to update the values of ψi→j as we go. If this procedure
converges it finds a solution to the BP equations. This iterative algorithm is known
as belief propagation and it works exactly (and efficiently) on trees and gives very
good heuristic results on large random graphs in sufficiently unfrustrated regimes.
There are as many BP equations as unknown cavity distributions and thus we
generically expect to find discrete solutions.2 However, there can be more than one
solution. For instance in the low temperature phase of the Ising ferromagnet, there
are three: the (unstable) paramagnet and the two (symmetry related) magnetized
solutions. For spin glass models with quenched disorder, there may be exponentially
many solutions, each corresponding to a macroscopically distinct magnetization pattern in the system. When this occurs, the belief propagation algorithm may fail
to converge – and for such systems, a thermodynamic analysis using the RS selfconsistency equation (4.12) is incorrect. In this case, one needs to take into account
the presence of multiple solutions of the BP equations, which can be done statistically
using an algorithm known as Survey Propagation and in the thermodynamic limit
using the ‘replica symmetry breaking’ cavity equations, which arise as a hierarchy of
distributional equations. These equations describe the statistics of solutions of the
BP equations. Although these analyses are important for a correct understanding
of many types of glassy optimization problems on tree-like graphs [87], we will not
consider them further in this thesis. We note that the jargon of ‘replica symmetry
breaking’ arises in a completely different approach to solving mean field glasses based
2
At least, if the degrees of freedom are not themselves functions, as in some of the quantum
models we will consider in later chapters.
93
on the so-called replica trick and Parisi ansatz. The terms have no intrinsic meaning
in the context of the cavity approach.
4.5
Quantum cavity method
The general ideology of the classical cavity approach is to find a relationship between
the cavity distribution ψi (σi ) of a local degree of freedom and the cavity distributions
of its neighbors. Formally, this relies on the ability to rearrange the partition sum in
terms of cavity distributions as follows:
Z =
X
e−β
P
hiji
Hij (σi ,σj )
=
=
Y X
0
e−βH0i (σ0 ,σi ) ψi→0
(σi )
(4.21)
σ0 i∈∂0 σi
{σi }
0
ψi→0
(σi )
XYX
0
e−βHij (σi ,σj ) ψj→i
(σj )
(4.22)
j∈∂i/0 σj
0
(σi ) is the (unnormalized) cavity distribution for the spin σi in the
where the ψi→0
absence of σ0 , and so on. In the previous section, we illustrated this approach using
a simple Ising model where the degrees of freedom σ took only two values, but it is
clear that such an iterative evaluation of the partition sum is possible for any classical
degrees of freedom with statistical interactions living on a tree lattice.
The problem with the quantum partition function
Z = Tr e−β
P
ij
Hij
(4.23)
is that the terms Hij need not commute and thus the exponential of the sum is not
simply the product of the exponentials. The most straight forward way to devise a
cavity method for studying quantum models on trees is thus to make them classical,
a technique we introduced to the study of quantum spin glasses in [81] and which has
since been taken up in the study of a number of quantum models on trees [110, 73, 26,
94
80]. Any reasonable quantum-classical mapping which preserves the underlying tree
structure will do. In this thesis, we will consider the path integral representation of the
transverse field Ising model in imaginary time (Chapter 5), the similar path integral
representation of the quantum spherical model (Chapter 8), and, the coherent state
representation of the AKLT model as a classical vector antiferromagnet (Chapter 6).
We note that these mappings are well-known in finite dimensional models where the
former pair map d dimensional quantum models to d + 1 dimensional classical models
and the latter has the unusual feature of mapping d to d. For us, these simian-friendly
transformations map from ‘tree’ to ‘tree + 1’ and ‘tree’ to ‘tree.’
An alternative, somewhat more abstract approach to developing a quantum cavity
analysis has also been recently introduced by several groups [61, 83]. Rather than
mapping the quantum model into a related classical model and then using a classical
analysis, these work directly in the quantum language by representing states of the
system and ‘cavity states’ of subsystems by appropriate reduced density matrices.
The cavity recursion relation is then an approximate recurrence on truncated reduced
density matrices rather than classical probability distributions and has the flavor of
a density-matrix renormalization group on a tree-like structure. We will not pursue
these approaches further in this thesis.
4.6
Population dynamics
In the RS cavity framework, all of the statistical observables of a model can be derived
from the cavity distribution P [ψ]. This distribution is the fixed point of the flow of
the cavity equation:
P [ψ] =
Z
q−1
Y
i=1
!
Dψi P [ψi ] hδ[ψ(σ) − ψ0 (σ; {ψi }, {J0i })]iJ
95
(4.24)
This equation must be solved for P [ψ(σ)], a distributions of distributions. In the study
of replica-symmetry breaking models, the 1RSB cavity equations that arise define
distributions of distributions of distributions. In a few examples, these distributional
equations may be solved analytically and it is often relatively straightforward to detect
the continuous onset of ordering by a stability analysis (see e.g., Chapter 7). Usually,
however, one must resort to numerical methods to solve such equations.
Fortunately, a straight-forward iterative technique exists for solving distributional
equations of the form (4.12): population dynamics [89, 87]. In this approach, we
represent a probability distribution such as P [ψ] by a population of NP representative
P
samples {ψj }N
j=1 . In order to find the distribution which satisfies Eq. (4.12), we
initialize the population in some fashion and then proceed schematically as follows:
1. Select q − 1 ψj from the current population randomly
2. Select q − 1 J0j randomly from P (J)
3. Merge the selected ψj using the couplings J0j to find a new distribution ψ0 .
4. Randomly replace one of the NP elements of the population with ψ0 .
5. Repeat until convergence (in some appropriate measure).
In practice, this procedure converges quickly in paramagnetic and ferromagnetic
phases but slows near phase transitions where the Lyapunov exponents near the
fixed points become very small. The accuracy of the representation of P [ψ] clearly
depends on the size of the population NP , much as a histogram improves with the
number of samples.
In the classical Ising case, ψj (σ) is naturally parameterized by a single real value,
the cavity field hj . More complicated models require more complicated parameterizations, as will be explored in some detail in Chapter 5. Indeed, it is even possible to
represent distributions ψ(σ) by populations of representative configurations of σ and
96
thus compute a population of populations. Or, in the treatment of 1RSB models, a
population of populations of populations! We will not here pursue such Malthusian
possibilities further.
97
Chapter 5
Transverse field Ising glass
In this chapter,1 we propose a generalization of the cavity method to the transverse
field Ising glass on fixed connectivity lattices. This a prototypical model for both
quantum spin glasses and adiabatic computation. The resulting equations describe
a Markov process for the on-site effective action whose stationary probability distribution can be found numerically using a population dynamics algorithm. This will
be the main result of the chapter. In this way we obtain the phase boundary in
the (Bt , T ) plane, the values of the usual thermodynamic quantities (free energy, energy, entropy and qEA , the Edward-Anderson order parameter [38]) and more typical
‘quantum’ quantities like the single spin von Neumann entropy. Indeed, from the fixed
point probability distributions for the effective actions we can calculate distributions
for all of the statistical properties of the system.
We note that the topic of quantum spin glasses on trees has been tackled before in
the literature from the point of view of statistical mechanics [71, 70] and computation
theory [18, 69]. However, in order to proceed analytically and due to the complexity
of the problem, these works have employed several uncontrolled approximations. We
will compare with them briefly in the core of the chapter. While we have used
approximations in our analysis as well, we believe they are better controlled, since we
1
Chapter based on work with A. Scardicchio and S. L. Sondhi [81]
98
show how to systematically improve them and how the results are robust with respect
to the improvements. More evolved computational methods for tackling very similar
problems have appeared in the literature on dynamical mean field theory [53] and
could possibly be applied to the quantum cavity method on glasses. These methods
may allow a study of the zero temperature case, which our approximations could not
capture.
5.1
Model and cavity solution
We consider the modification of the classical Ising spin glass Hamiltonian (4.1) due
to the introduction of a transverse magnetic field
H=−
X
(ij)
Jij σiz σjz − Bt
X
σix .
(5.1)
i
This is called the transverse field Ising spin glass. The Ising variables of the classical
model have been replaced by Pauli matrices σ z and the magnetic field couples to the
matrices σ x . The fact that σ z and σ x do not commute gives rise to a host of interesting
new features due to the interplay of quantum mechanics and disorder [45, 84, 100].
The usual Suzuki-Trotter decomposition allows us to rewrite the problem in terms
of Nt Ising spins per quantum spin, where the number Nt needs to be sent to infinity
eventually. The additional dimension which is introduced in this way is the usual
imaginary time. The σiz σjz interactions are time-translation invariant (the disorder
is correlated in the time direction) while the σ x terms give a ferromagnetic nearestneighbor interaction in the time direction. Before writing the Hamiltonian let us
introduce some notation.
For any finite Nt we will refer to the Ising spin configuration at a given site i as
a “rod” of spins. The rod at site i is described by Nt spins σi (t) where t takes values
from 0 to β in steps of ∆t = β/Nt , with periodic boundary conditions σ(0) = σ(β).
99
This notation is convenient if the limit Nt → ∞ is eventually performed, since the
rod is represented by a function σ(t) : [0, β] → {−1, 1} with σi (0) = σi (β). The rod
statistics are described by a probability distribution ψ[σ(t)], a functional of σ(t), that
gives a positive real number for every configuration σ(t). The normalization condition
P
reads {σ(t)} ψ[σ(t)] = 1.
The partition function is written as
Z=
X
e−βH[σ]
(5.2)
{σi (t)}
where the Hamiltonian is:
βH = −
XX
t
(ij)
∆tJij σi (t)σj (t) − Γ
XX
t
σi (t)σi (t + ∆t).
(5.3)
i
We can also write this as a sum over links of the energy of a link:
βHij = −
X
t
1 X
∆tJij σi (t)σj (t) − Γ
σi (t)σi (t + ∆t)
q
t
(5.4)
where a fraction 1/q of the imaginary-time interaction is associated to each link (there
are q links per spin). Here Γ = 21 ln coth(βBt /Nt ).
Notice that Γ > 0 so the system is ferromagnetic, and moreover when βBt /Nt 1
then Γ 1 and the coupling along the time direction is strongly ferromagnetic. In
particular, for Bt = 0 the spins in any given rod are locked together as Nt useless
copies of a single Ising spin. Thus, the results reduce to the classical case smoothly.
The spatial tree-like structure of the original problem is reflected in the tree-like
structure of the interaction between rods. We can therefore imagine an iteration
process with rods replacing the spins, in which we have q − 1 cavity rods ψi [σi (t)]
which are merged and determine the state of the rod ψ0 [σ0 (t)] (see Fig. 5.1). This
corresponds to a recursion relation for the calculation of the partition function of the
100
branches, analogous to the classical equation (4.6):
1
ψ0 [σ0 (t)] =
Z
X
e
−
Pq−1
i=1
βH0i
{σi≥1 (t)}
q−1
Y
ψi [σi (t)].
(5.5)
i=1
Iterate
Figure 5.1: Iteration of cavity rods. There are periodic boundary conditions in the
imaginary time (vertical) direction. Other cavity operations are analogous.
Just as in the classical case, this iteration equation is already enough to solve an
interesting ferromagnetic problem. Consider the case q = 2 and Jij = J. For q = 2
our Bethe lattice is a simple chain of spins, moreover all J’s being equal we can take
J > 0 without loss of generality. We then recover the well known ferromagnetic Ising
chain with a transverse field, an exactly solvable system. One way to solve it is to
use Onsager’s transfer matrix method. In fact, the iteration equation (5.5) can be
rewritten as just such a transfer matrix equation where ψ is the 2N dimensional vector
and e−βH01 the transfer matrix T :
Zψ0 = T · ψ1 .
(5.6)
The fixed point of this iteration gives the eigenvector ψ corresponding to the largest
eigenvalue Z. In the limit Nt → ∞, this is an exact solution which contains all the
information about classical and quantum phase transitions.
101
Returning to the case of the spin glass, we can write down the quantum cavity
fixed point equation analogous to the classical equation (4.11) immediately:
PF P [ψ[σ(t)]] =
=
δ ψ[σ(t)] − ψ0 [σ(t); {J0i , ψi }q−1
]
i=1
J0i ,ψi
R Qq−1
i=1
Dψi PF P [ψi ] dJ0i P (J0i ) δ ψ[σ(t)] − ψ0 [σ(t); {J0i , ψi }q−1
i=1 ]
(5.7)
where the iterated rod action ψ0 [σ(t); {J0i , ψi }q−1
i=1 ] is given by Eq. (5.5) and P (J0i )
is the fixed prior distribution for couplings Eq. (4.10). This is a functional equation
for PF P [ψ[σ(t)]], the fixed point probability distribution of the effective distribution
describing iterated cavity rods. In the limit Nt → ∞, it is exact but difficult to solve
in closed form. It is certainly possible that analytic progress can be made, but we
have not succeeded thus far. However, it is amenable to numerical study at finite Nt
under certain approximations and also perhaps by continuous time Monte Carlo for
Nt → ∞. In the remainder of this section we will explore the finite Nt approach.
We must first parameterize our generic vector ψ in the 2Nt -dimensional space of
the configurations of the rods. In principle it is described by 2Nt − 1 real numbers,
which can be reduced by a factor O(Nt ) by exploiting time-translation symmetry and
the periodic boundary conditions. A natural way to parameterize it is in term of the
effective action of the rod
ψ[σ(t)] = e−S[σ]
(5.8)
where we expand S in a series of increasing clusters of interacting spins.
S[σ] = − log Z − h∆t
−
X
t,t0 ,t00
X
t
σ(t) −
X
t,t0
∆t2 C (2) (t0 − t)σ(t)σ(t0 ) −
∆t3 C (3) (t0 − t, t00 − t0 )σ(t)σ(t0 )σ(t00 ) + ...
(5.9)
In principle, the sum includes up to Nt -spin interaction terms (the normalization
factor has been included as a spin-independent term in the effective action). In
102
practice, we truncate the action expansion at second order to keep the numerical
requirements manageable. We comment below on the limits in which this truncation
is exact. Notice that C (2) (t) is the kernel for 2-point in time interactions, not the
dynamical two-point correlation function, often denoted c(2) (t) = hσ(t)σ(0)i.
The functions h, C (i) are random quantities characterized by the Markov process
defined by the iteration procedure. By writing the representations of the vectors ψ in
terms of the effective action (5.9) we can rewrite the iteration equation as
e
−S[σ,{h0 ,C0 }]
=
X
e
−
Pq−1
i=1
βH0i
{σ1 (t)},...,{σq−1 (t)}
q−1
Y
e−S[σ,{hj ,Cj }] .
(5.10)
j=1
(2)
(3)
This gives an implicit update map from the ‘old’ q − 1 parameters hj , Cj , Cj , ...
(2)
(3)
and the couplings J0j , Bt to the ‘new’ parameters h0 , C0 , C0 , .... The statistics
generated by this Markov process and in particular its fixed point distribution
P (h, C (2) , C (3) , ...)
(5.11)
are the solution of the problem.
In the next section we show how a simple minded Trotter discretization with Nt
relatively small (Nt = 6 to 11) delivers a great deal of information about the quantum
spin glass phase.
5.2
Numerical approach
We solve the fixed point equation (5.7) numerically using a population dynamics
algorithm analogous to that of Mézard and Parisi[89, 88]. We represent P [ψ] by a
finite population of Nrods rod actions {ψi [σ(t)] = e−Si [σ(t)] }, where the expansion (5.9)
of Si is truncated to second order. Each rod action is therefore specified by 1 + b N2t c
distinct numbers (h, C (2) (t)) after exploiting the periodicity of imaginary time.
103
This population is initialized from an appropriate uniform distribution and then
iterated as follows:
1. Select q − 1 rods ψi randomly from the population and q − 1 random J0i .
2. Use (5.5) to calculate the effective action on an iterated cavity spin ψ0 from
these rods. In principle, higher order interactions may be generated but we
truncate them by finding the second order action that exactly reproduces the
free energies of a series of domain wall configurations of varying width.
3. Randomly replace one element of the population with ψ0 .
4. Repeat until convergence in some measure of the population, for example the
order parameter qEA .
In practice, this procedure converges quickly deep in either the glassy or paramagnetic
phase but slows near the phase transition as the flow of P [ψ] under the iteration
equation near the paramagnetic fixed point becomes marginal.
Given PF P [ψ], we can calculate the sample averaged free energy density and local observables such as the link energy, site magnetization, Edwards-Anderson order
parameter qEA = hσi i2T i and reduced von Neumann entropy SvonN = −trρ0 log2 (ρ0 )
by standard Monte Carlo sampling of these quantities. The free energy density is
given by equation (4.16). The reduced entropy and transverse magnetization may be
derived in the usual way from the reduced density matrix ρ0 for a spin. We calculate
this by performing a merge operation onto a “broken rod” (see Fig. 5.2), in which
periodic boundary conditions are not enforced. The various elements of the reduced
density matrix correspond to imposing different values on the top and bottom spin
of the broken rod and summing out all other spins in the partition function. Finally,
calculating the average internal energy can be done by averaging the Hamiltonian
(5.1).
104
ψ1
ψ2
Figure 5.2: Merging of two cavity rods onto a “broken” central rod. Here Nt = 3
but instead of imposing periodic boundary conditions everywhere we depict σ(0) and
σ(Nt ) as independent spins. Each vertical link corresponds to a Γ coupling while each
horizontal link corresponds to a Jij /2Nt coupling. The dashed lines connecting the
cavity rods indicate identification of the top and bottom spins for each rod and the
wiggly line indicates an effective action. By imposing different values on the top and
bottom spin of the broken rod and summing out the rest, the elements of the reduced
density matrix ρ0 may be determined.
5.2.1
Approximations
Action representation
By parameterizing the quantum dynamics through the action expansion (5.9) we do
not make any a priori assumptions about the nature of the spin-spin correlations
in the time direction – that is, if we could keep all of the terms in the expansion,
it would be an exact treatment. We avoid in this way the spherical approximation
which has been used in Ref. [71, 70], since the results thus obtained do not reduce to
the well-known classical results for Bt = 0.
In practice, however, we truncate the cavity actions to second order. Most usefully,
this corresponds to the leading order term in a large connectivity expansion of the
effective action. Indeed, in a large q treatment, in which the couplings Jij must be
√
scaled as 1/ q for the disordered model, one finds that the one- and two-body terms
in the effective action for the rod at the root of a tree are O (1/q 0 ), the three- and
four-body terms are O (1/q 1 ) and so on[53]. The truncation to second order is thus
both exact and necessary in the q → ∞ limit. We note that the oft-used static
approximation[113, 116] is thus incorrect even at infinite connectivity for disordered
105
models. The truncation to second order is also exact at high temperature, regardless
of the value of q.
Numerical investigations of small systems suggest that the higher order interactions are quantitatively small more generally, even at q = 3. This is especially true
at small Bt , where the strong ferromagnetism in the time-like nearest neighbor bonds
dominates. We note that there is also some error in the numerical fitting of an iterated cavity distribution to a truncated action. This problem is reminiscent of a
maximum entropy model for the statistics of signals and the inverse Ising problem of
computer science [62]. However, even in the disordered system, the interaction of the
spins along imaginary time is always ferromagnetic and the problem does not present
the difficulties that usually accompany the inverse problems in general statistical mechanics. In other words, although the original problem is fundamentally frustrated,
none of this frustration appears in the single spin dynamics. The frustration is taken
care of in the treatment of the relevant parameters h, C (i) as random numbers.
For comparison, we note that our approach is closely related to self-consistent
dynamical mean field theory (DMFT) methods, which also truncate to non-local two
point interactions in the single-spin effective action (these are the so-called Weiss
functions or bare Green’s functions of DMFT treatments). However, DMFT techniques cannot be used straightforwardly for disordered systems. One cannot assume
that the imaginary-time propagator (or action) for a spin is equal to that of the spins
which surround it but rather that they come from the same probability distribution.
If this observation is taken into account then we expect to recover functional equations for the distribution of single site Green’s functions analogous to those we have
proposed for the single site action.
106
Finite discretization error
Finally, we comment on where in the (τ, Bt ) phase diagram the fixed Nt approach
will give reliable results. As mentioned above, on the classical line Bt = 0, all spins in
a rod are locked together by divergent nearest neighbor interactions and the Trotter
decomposed system (for any Nt ) reduces to the classical system exactly. In the
opposite limit of large Bt at fixed Nt , the planes of the Trotter system decouple as
Γ → 0. Each of these planes is an exact copy of the original Bt = 0 spin glass model
except with couplings Jij /Nt . These will therefore undergo independent thermal phase
transitions at τ = τc /Nt and no τ = 0 critical field will be detected. This explains
the phenomenon of “asymptotic critical lines” that we note in our finite Nt phase
diagrams.
More generally, for the finite Nt approximation to be valid, Nt > ktyp where ktyp
is the typical number of kinks in a rod. For a single spin in a transverse field, a
straightforward calculation shows hki = 2βBt tanh βBt , which for large βBt reduces
to hki ≈ 2βBt . Within the paramagnetic phase this calculation remains nearly exact,
although in the spin glass phase hki will be suppressed by the presence of longitudinal
cavity fields. However, this inequality Nt > 2βBt remains a good indicator of the
quality of the approximation and agrees with the regions where it clearly breaks down.
Thus, this expansion is particularly useful close to the line Bt = 0, and we will
see that it gives reasonable results and insights into the structure of the problem also
deep in the quantum spin glass phase. On the other hand, the other region of interest
T = 0, Bt ' Btcrit should not be addressed with this expansion, unless the description
of the spins in terms of continuous time functions σ(t) turns out to be treatable in
the future. In this chapter, we will not be able to make definitive statements about
the nature of the quantum phase transition which occurs at this point but we are
definitely able to make statements about the nature of the spin glass phase when
quantum effects are not negligible.
107
5.3
Numerical results
5.3.1
Phase diagram
We present numerical results for an investigation of the q = 3 connectivity model
using a naive (exact) approach to the exponential summation involved in the cavity
iteration and merging operations.2 Fig. 5.3(a) shows the phase diagram calculated
at Nt = 10, Nrods = 2500, Niter ∼ 1000Nrods and suggestively fit to Nt → ∞ using
asymptotic expansions in 1/Nt2 . Qualitatively, all is as might be expected:
• At any Nt , the phase transition curve predicts a Bt = 0 critical temperature in
√
agreement with the analytic prediction of τc = 1/ tanh−1 (1/ q − 1) ≈ 1.13.
• The upturn in the Nt = 10 phase boundary at low temperature is due to the
finite discretization of time, which leads to an asymptotic phase transition line
at τ =
τc
Nt
≈ 0.113.
• While the fits to Nt → ∞ are certainly approximate, we believe that the true
τ = 0 critical field lies between 1.5 and 2. We believe continuous time techniques
will allow dramatic refinement of this estimate and investigation of the quantum
critical region.
Our phase diagram clearly disagrees with that of Kopec and Usadel[71, 70], who
treat the identical model using a soft spherical approximation and find that both
the critical temperature and critical transverse field are depressed relative to our
values. Presumably this suppression of ordering arises due to the stronger effect of
fluctuations in the softened model.
2
We note that a rearrangement of the sum over configurations according to the number of kinks
looks like a natural perturbative expansion in Bt for the system and naively scales polynomially
O(Ntk ) where k is the maximum number of kinks summed. Unfortunately, little is gained from
this rearrangement in terms of the accurately computable region of the phase diagram: the expected
number of kinks for relevant rod configurations scales ∼ 2Bt /τ and thus the sum is again exponential
in Bt /τ .
108
Figure 5.3(b) shows the instance averaged single site von Neumann entropy SvonN
which has a remarkably clear maximum near the phase transition curve above the
classical line. This reflects the strength of quantum correlations even at the finite
temperature phase transition. See Ref. [10] for discussion of local measures of entanglement at finite temperature.
Zooming in on the horizontal stripe at Bt = 1 indicated on the phase diagram,
we find that qEA vanishes linearly at the critical temperature (Fig. 5.4(a)). This
reflects the underlying broadening transition in P [ψ], which can be seen sharply in
the variances of each of the effective action coefficients (Figs. 5.4(b,c)). We use this
behaviour to estimate sharp transition points despite softening due to critical slowing
in the convergence of our procedure.
Finally, we note that much of the phase diagram is surprisingly stable to variation
in Nt . We have explored various regions of the phase space at Nt = 6, 7, 8, 9, 10, 11.
The classical line (Bt = 0) at all temperatures is completely stable down to Nt = 1
as expected. Perhaps more surprisingly, moving between Nt = 8 and Nt = 10, qEA
is essentially stable below Bt = 1 down to temperatures τ ∼ 0.15. Of course, the
high field, low temperature part of the phase transition curve moves downward as
the finite discretization asymptote goes towards the τ = 0 axis. See Fig. 5.5 for the
low temperature critical curves estimated using vertical stripes run at five different
temperatures (corresponding to β = 3.5, 4, 4.5, 5, 5.5) at various Nt .
5.3.2
Structure of the glassy phase
Since the order parameter is a probability distribution of an action, there is a rich
structure to be investigated at even a single point (τ, Bt ) within the glassy phase. In
Fig. 5.6, the marginal probability distribution of the field term h in the cavity action
is shown at the four points indicated on Fig. 5.3(a). The two lower distributions
lie on the classical line (Bt = 0), one deep within the glassy phase and one near
109
2.5
1
0.9
3
Nt = 10
2
PM
0.8
2.5
0.7
1.5
2
t
1
1.5
0.5
1
0.4
SG
0.5
0.2
0
0
0.5
τ
0.3
0.5
Critical Asymptote (Nt=10)
0
0.6
Bt
B
t
Fit N → ∞
1
1.5
0
0.5
1
1.5
τ
Figure 5.3: (a) Phase diagram at q = 3. The solid phase transition curve has been
calculated at Nt = 10, Nrods = 2500, Niter = 1000Nrods on a fine mesh in the (τ, Bt )
plane. The vertical dotted line is the asymptotic critical line for large Bt at Nt = 10
(i.e. τ = τc /Nt ). The points marked x with error bars indicate Nt → ∞ fits based
on Fig. 5.5. The dashed transition curve is a weighted quadratic fit through the
estimated low temperature points and the Nt = 10 points in the range 0.5 < τ < 1.
This leads to an estimated Btc = 1.775 ± 0.03. As this fit is clearly heuristic, we have
suggested a much larger range for our estimate of Btc in the Figure. The stars and
stripes indicate points in the phase space which we have investigated in more detail
below. (b) The average von Neumann entropy SvonN (in bits) of a central spin as
a function of (τ, Bt ) at Nt = 8. The dashed line indicates the estimated region of
validity of the discretization approximation (Bt ≤ Nt τ /2).
3
2
0.1
V ar(C(1))
qEA
2
V ar(h)
1
0.05
1
V ar(C(2))
0
0
τc = 0.921
-0.05
0
-1
τc = 0.913
-1
τc = 0.915
-2
0.8
0.9
τ
1
0.8
0.9
τ
1
0.8
0.9
1
τ
Figure 5.4: These graphs all correspond to the horizontal slice at Bt = 1 shown in
Fig. 5.3. The order parameter qEA and all effective action variances undergo mean
field like transitions at the critical temperature (e.g.. qEA ∼ |τ − τc |1 ). This allows us
to estimate the critical temperature precisely despite softening due to critical slowing
down near the phase transition. These curves were calculated at Nt = 8. However,
the results are stable to increasing Nt to 10 to within an error of ±0.005 in τc .
110
3.2
N =6
t
3
Nt = 7
2.8
Nt = 8
Nt = 9
Bct
2.6
Nt = 10
Nt = 11
2.4
Nt = ∞
2.2
2
1.8
1.6
0
0.05
0.1
0.15
τ
0.2
0.25
0.3
Figure 5.5: The phase transition curve in the high field, low temperature regime
at various Nt . The vertical dashed lines indicate the finite discretization critical
asymptotes. The estimated curve for Nt → ∞ in Fig. 5.3a is given by fitting Bt (Nt ) =
a/Nt2 + b/Nt4 + c to the points calculated at each temperature and then sending Nt
to infinity in the result. These fits suffer from a paucity of data, but are suggestive
nonetheless.
the transition. It is clear that the distinctive features of the classical solution are
reproduced here: a Gaussian-like structure around h = 0 near the phase transition
with the appearance of delta function spikes on the integer fields deep within the
phase. At Bt = 1, the qualitative picture of spread from narrow Gaussian near the
phase transition to broader, bumpier distribution remains. It is less clear whether
the sharply defined spikes on integer fields would remain at τ = 0 with large Bt .
Further structure can be found in the nontrivial probability distribution for the
interaction terms that develop in the spin glass phase. Figure 5.7 shows the histogram
for various marginal and h-conditioned distributions of the nearest neighbor and next
nearest neighbor interactions terms at (τ = 0.25, Bt = 1) (cf. Fig. 5.6(top left)). We
can qualitatively understand many features of these distributions:
• The two-spin interactions are ferromagnetic and the effect of coupling to neighboring rods is only to enhance the ferromagnetic interaction from the bare near-
111
est neighbor interaction on a single rod (Γ). Indeed, this Γ sets the minimum
strength of C(δT ), as can be seen in the top row.
• The strength of two-spin interactions are strongly anticorrelated with the
strength of the cavity field h as can be seen from the decomposition of the full
marginal distributions of C(δt) and C(2δt) into the small (middle column) and
large (right column) cavity field conditioned distributions. Large cavity fields
on a central spin come from large fields biasing neighboring rods. These fields
pin the neighboring spins more strongly and reduce the ability of those spins
to mediate interactions in time between the central rod spins, reducing the
effective two-spin interaction.
• The multimodal spikiness in these distributions reflect the spikiness in the low
temperature cavity field distributions through the field-interaction correlation.
Unfortunately, we have yet to develop a more significant analytic understanding of
these correlations nor a means to extrapolate them to zero temperature in the presence
of the transverse field.
Finally, we emphasize that the phase transition is signaled by a singular broadening of P rather than any singularity in its first moments or in the structure of the
typical imaginary time action. This can be seen in the smooth evolution of hC(2∆t)iJ
through the phase transition in Fig. 5.8a and the similarly smooth evolution of the
average two point correlation function hσ(0)σ(t)iJ . This is in contrast to the ferromagnetic case in which the distribution P (ψ) would exhibit spontaneous symmetry
breaking but remain deterministic.
5.4
Discussion and further work
The cavity method has a long and illustrious history in the study of statistical systems – from Bethe’s early work on the Ising ferromagnet to modern studies of random
112
Figure 5.6: The distribution of the field term of the cavity rod action at the four
different points in phase space labeled by the stars on Fig. 5.3. The distinctive distribution of the low temperature replica symmetric classical spin glass is reproduced in
the bottom left corner while the three other points all lie closer to the phase boundary
in the τ or Bt directions.
constraint satisfaction problems in computer science. In this chapter, we have introduced a new variant for studying disordered quantum systems within an imaginary
time formalism. This represents an intuitively appealing, natural synthesis of the
classical disordered model techniques with the quantum homogeneous models studied in DMFT. We note that our framework can be simply adapted to study many
other transverse field Ising systems on trees with fixed or fluctuating connectivity –
such as the ferromagnet, diluted ferromagnet or biased glass.
We have shown that the transverse field Ising glass on a Bethe lattice of connectivity three has a phase transition line all the way from the classical Bt = 0, T = 1.13
to the quantum Bt ≈ 1.75, T = 0. At finite temperature, the transition is classical
113
Figure 5.7: Histograms of nearest neighbor (top row) and next nearest neighbor
(bottom row) interactions in imaginary time at τ = 0.25, Bt = 1 (in the SG phase).
The second and third column provide the conditional distribution of the interactions
given |h| is small or large.
and mean field like. Inside the frozen phase the picture is similar to the classical
case: when a randomly chosen spin is extracted from the graph, its effective action
(analogous to the cavity field) is well-defined in the paramagnetic region but is a
random functional in the spin-glass phase. In this phase all of the local observables,
classical and quantum, are therefore also random variables. In principle one could
also study the properties of the entanglement of distant spins, which cannot be done
in the better known but fully connected SK model.
The reader familiar with classical spin glass theory will have noticed our avoidance
of the important issue of replica symmetry breaking (RSB), which is widely believed
to be a feature of a correct treatment of the random graph Bethe lattice. In this
114
Figure 5.8: (a) The average next to nearest neighbor interaction in time C (2) (2∆t)
of a cavity rod action at Bt = 1, varying the temperature. The bars indicate the
variance of C (2) (2∆t). Notice the zero variance above the critical temperature. (b)
Single site imaginary time correlation function hσ(0)σ(t)iJ at τ = 0.25 for various Bt
passing through the transition at Btc ∼ 1.85.
connection, we note that our simpler treatment is indeed correct for models on Cayley
trees with fixed boundary conditions. Quantum RSB phenomena certainly deserve
further study: the conceptual difficulties of the RSB ansatz become even thornier in
the presence of quantum tunneling. We note that formally breaking replica symmetry
at the one step level (1RSB) should be straightforward in our framework. In analogy
with Ref. [89, 88], one should introduce populations of populations of effective actions
and weigh them according to their free energy (as one does for different solutions
of the TAP equations). This straightforward modification of the algorithm makes it
computationally considerably more time consuming. For this first pass, we decided
not to embark on such a project.
Unfortunately, we are not aware of Quantum Monte Carlo or other numerical
studies on the transverse Ising model on the Bethe lattice with which to compare our
results. The primary difficulty that has prevented the direct simulation of this system
is that it requires a very large number of spins to approximate the infinite system
effectively. Since a random graph with fixed connectivity has an extensive number of
115
loops of lengths ` ∼ O(ln N/ ln(q − 1)), N needs to be exponentially larger than any
statistically relevant length scale (e.g. the coherence length).
Another direction for further development would be to find a spin glass (or otherwise) model amenable to analytic treatment within the quantum cavity method. A
“soft spin” Gaussian model would do, since the path integrals to be performed in an
iteration could be computed exactly. However, we do not believe this model has a
spin glass phase in the absence of higher order couplings. Whether one could treat
such a coupling perturbatively is a question worthy of further exploration.
116
Chapter 6
AKLT models with quantum spin
glass ground states
6.1
Introduction
The study of quantum antiferromagnets has proven among the most enduring themes
in modern condensed matter physics.1 The interplay between frustration and quantum fluctuations leads such systems to exhibit a variety of interesting ground states.
In this context the lattice models constructed by Affleck et. al [3, 2] are particularly
useful for they build in a great deal of both these effects using simple local projectors, which allow their ground states to be determined analytically. These AKLT
models have spins given by S = z2 M , where M is a positive integer, and z the lattice coordination number. In principle, they may be defined on any graph, but in
practice one usually maintains fixed connectivity in order to have the same spin on
each site. The associated ground states have the added feature that their wavefunctions can be written in Jastrow (pair product) form, which allows us to view their
ground-state probability densities as Boltzmann weights corresponding to a nearest
1
Chapter based on work with S. A. Parameswaran, S. L. Sondhi and F. Zamponi [80].
117
neighbor Hamiltonian for classical vector spins on the same lattice. Using this unusual quantum-classical equivalence one can understand many properties of the states
by studying the associated classical model.
The initial construction of the AKLT models was motivated by the search for
quantum disordered states in low dimensions. This works only too well: in d = 1
and d = 2 the mapping to finite temperature classical models discussed above ensures, by the Mermin-Wagner theorem, that all cases lead to quantum paramagnetic
or valence bond solid ground states. In d > 2 this is no longer true and a computation is needed to decide which models order and which do not. In a recent paper,
Parameswaran, Sondhi and Arovas [103] showed via Monte Carlo simulations and
mean-field arguments that AKLT models on the diamond and pyrochlore lattices exhibit quantum-disordered ground states for small spin sizes while on the cubic lattice
all models exhibit Néel order.
In this chapter we take this exploration of higher dimensional AKLT models in a
different direction—we study them on locally tree-like lattices of fixed connectivity z,
which are known to physicists as Bethe lattices. Here we shall consider two physically
distinct systems. The first, is the Bethe lattice constructed as the limit of a family of
Cayley trees. This construction yields a system with a finite surface to volume ratio
and without loops. The second is a typical member of the ensemble of random graphs
of fixed connectivity. These graphs are locally tree-like in the thermodynamic limit;
however they also have long loops of logarithmically divergent size. These loops of
both even and odd lengths introduce topological frustration into the system. The two
constructions of locally tree-like lattices yield different physics.
For the infinite Cayley tree, we exhibit an exact solution using the quantumclassical correspondence. Specifically, we use a generalized transfer matrix technique
to obtain exact solutions for various statistical quantities in the ground state of the
tree. We note that the AKLT model on the Bethe lattice has been studied before
118
directly within the quantum formalism [42]; we suspect that readers will find our
solution simpler. We find one quantum disordered state (M = 1 on the z = 3 tree)
and two that are critical (M = 2 on z = 3 and M = 1 on z = 4), in that the
correlation functions decay exponentially at precisely the rate required to balance
the exponential growth of the graph. All other cases exhibit Néel order. We address
the question of whether the bulk excitations are gapless in cases when the AKLT
wavefunction has critical or Néel correlations. We find, perhaps surprisingly, that the
system is always gapped to local excitations, and that the only gapless excitation is a
global one connecting the different broken-symmetry ground states. We connect this
to related work in Ref. [79], in which we conjecture that this is a generic feature of
symmetry breaking quantum models on the Bethe lattice, related to the spectrum of
the graph Laplacian.
On random graphs of fixed connectivity, Néel ordering in the companion classical
model is frustrated by the presence of the long loops. To study this case, we appeal to
the cavity techniques familiar from the theory of classical disordered systems. These
have been applied recently to a variety of discrete statistical mechanical problems on
random graphs and there is much evidence that the (approximate) techniques are on
solid ground. From this analysis, we conclude that for z ≤ 10 there are disordered
states at small spin and spin glass ground states at larger spin as well as a couple of
cases where the state is critical. For z > 10 all AKLT models have ground states with
spin glass order. By spin glass order, we mean states with fixed but randomly oriented
local magnetizations and that the set of such states is larger than those connected
by global rotations alone. We argue that the spectrum of local excitations above the
pure states in this set is again gapped.
Of our various results we would especially like to flag these last mentioned. The
nature of quantum glass phases is a subject of much interest – especially as to how
much of the elaborate framework of the classical subject may be lifted into the quan119
tum world. The AKLT construction provides a direct line of approach to this problem
and does so using Hamiltonians without random couplings but from graph disorder
alone.
This chapter is organized as follows: in Sec. 6.2, we introduce the AKLT model
on an arbitrary graph via the Schwinger boson formalism. We proceed to construct
a companion classical model that captures the structure of the ground state wavefunction by introducing a basis of SU (2) coherent states. In Sec. 6.3, we develop
transfer-matrix technology to solve the companion classical model on the (bipartite)
Bethe lattice exactly, and obtain the transition temperature and correlation functions
in the paramagnetic and Néel-ordered phases. In Sec. 6.4 we investigate the energy
gap using a variational ansatz for the excited states. Finally, in Sec. 6.5 we consider
the extension of this analysis to the spin glass transition expected on regular random
graphs and consider some of the quantum consequences of the classical glassy phase.
6.2
AKLT states: a brief review
The central idea of the AKLT approach [2] is to use quantum singlets to construct correlated quantum-disordered wavefunctions, which are eigenstates of local projection
operators. One can then produce many-body Hamiltonians using projectors that extinguish the state, thereby rendering the parent wavefunction an exact ground state,
often with a gap to low-lying excitations. A general member of the family of valence
bond solid (AKLT) states can be written compactly in terms of Schwinger bosons
[12]:
|Ψ(M )i =
Y
b†i↑ b†j↓
hiji
−
b†i↓ b†j↑
M
|0i .
(6.1)
This assigns M singlet creation operators to each link hiji of an underlying lattice.
The total boson occupancy per site is given by zM , where z is the lattice coordination number, and the resultant spin on each site is given by S = 21 zM . Given any
120
regular graph, the above construction defines a family of AKLT states labeled by the
size of their spins S = 21 zM . For more details regarding this construction and the
corresponding Hamiltonians, see Ref. [103].
The AKLT states have a convenient representation in terms of SU (2) coherent
states, as first shown in Ref. [12]. In terms of the Schwinger bosons, the normalized
spin-S coherent state is given by |n̂i = (p !)−1/2 (zµ b†µ )p | 0 i, where p = 2S, with
z = (u , v) a CP1 spinor, with u = cos(θ/2) and v = sin(θ/2) eiϕ . The unit vector
n̂ is given by na = z † σ a z, where ~σ are the Pauli matrices. In the coherent state
representation, the general AKLT state wavefunction is the pair product Ψ({n̂i }) =
Q
h{n̂i }|Ψi = hiji (ui vj − vi uj )M . Following Ref. [12], we may write |Ψ({n̂i })|2 ≡
exp(−βHcl ) as the Boltzmann weight for a classical O(3) model with Hamiltonian
X 1 − n̂i · n̂j ,
Hcl = −
ln
2
(6.2)
hiji
at inverse temperature β = M . All equal time quantum correlations in the state |Ψi
may then be expressed as classical, finite temperature correlations of the Hamiltonian
Hcl .
Some immediate consequences of this quantum-to-classical equivalence were noted
in Ref. [12]. On one and two-dimensional lattices, the Hohenberg-Mermin-Wagner
theorem precludes long-ranged order at any finite value of the discrete quantum parameter M . In three dimensions, there is no a priori reason to rule out long-range
order. In fact, as shown in Ref. [103], the simple cubic lattice has no quantumdisordered states at any M , while the diamond lattice has a single such state for
M = 1; on frustrated lattices such as the pyrochlore, such states are believed to exist
for many values of M .
As is evident from (6.1), we may define the AKLT states on an arbitrary graph;
if the graph has fixed connectivity z, then the resulting model has the same spin on
121
each site. On graphs with a boundary, this is not automatic, since the boundary sites
will have fewer neighbors z 0 . There are several ways to deal with this boundary effect.
The first is to work with a system with a lower spin on the boundary: in that case
S 0 = 12 z 0 M . The quantum state of this non-homogeneous system is unique. Another
option is to add (z − z 0 ) additional Schwinger bosons of either flavor to the edge sites;
there is not a unique way in which to do this, leading to a multitude of degenerate
ground states classified by the state of the boundary spins. When translated to the
companion classical model, the latter option can be viewed as connecting each of
the boundary spins to (z − z 0 ) fixed spins, each with an orientation specified by the
behavior of an independent spin-1/2 degree of freedom; thus, the different degenerate
states of the homogeneous AKLT model on a graph with boundary can be understood
by choosing different fixed boundary conditions for spins in an additional, outer ring
of leaf spins. Finally, one can opt to get rid of the boundary by, for instance, taking
periodic boundary conditions on a Euclidean lattice. As we will discuss in Sec. 6.5,
when generalized to tree-like graphs, this approach leads to a spin glass phase in
the companion model and thus provides a non-trivial new quantum spin glass to the
AKLT phase diagram.
6.3
Transfer matrix solution of the classical problem on trees
The classical Hamiltonian that describes the properties of the ground-state wavefunction of an AKLT model with singlet index M , written in the basis of SU (2) coherent
states is given by
βHcl = −M
X
hi,ji
1 − n̂i · n̂j
log
2
122
(6.3)
Before we proceed, we note that on bipartite graphs we can perform a gauge
transformation by flipping every spin at odd depth to obtain a ferromagnetic model.
This gives us
βHcl = −M
X
hi,ji
1 + n̂i · n̂j
log
2
(6.4)
As usual in the treatment of tree models, we consider the statistical state ψ 0 (n̂0 )
(marginal distribution) of a cavity spin n̂0 at the root of a branch of the tree. This
unnormalized distribution can be found in terms of the cavity states of its z − 1
neighbors by summation:
0
ψ (n̂0 ) =
=
Z
Z
Dn̂1 · · · Dn̂z−1 T (n̂0 , n̂1 )ψ 1 (n̂1 ) · · · T (n̂0 , n̂z−1 )ψ z−1 (n̂z−1 )
Dn̂01 · · · n̂0z−1 M (n̂0 ; n̂01 , · · · n̂0z−1 )
(6.5)
Z
× Dn̂1 · · · Dn̂z−1 T (n̂01 , n̂1 )ψ 1 (n̂1 ) · · · T (n̂0z−1 , n̂z−1 )ψ z−1 (n̂z−1 )
where
T (n̂0 , n̂1 ) = e
β log[
1+n̂0 ·n̂1
2
]=
1 + n̂0 · n̂1
2
β
(6.6)
is the transfer matrix of the AKLT model and
M (n̂0 ; n̂01 , · · · n̂0z−1 ) = δ(n̂0 − n̂01 ) · · · δ(n̂0 − n̂0z−1 )
(6.7)
is the merge matrix.
The merge matrix M defines a multilinear map from the z − 1 state spaces of the
neighbor spins to the state space of the root. This lifts naturally to the appropriate
complexified tensor product spaces2 and thus we will find it natural to write the merge
2
A few comments on the nature of the classical statistical state space of a vector spin are in order,
as we have so cavalierly complexified and tensored it into a much more quantum mechanical looking
system. Physical cavity distributions ψ(n̂) must be real, normalizable, nonnegative functions on the
sphere. By its construction as a marginalization (summing out) procedure, equation (6.10) must
produce such a physical output given physical inputs, even though we have extended it over C. A
more important subtlety arises in the normalization of states – the standard L2 norm associated
123
and transfer operations abstractly using Dirac notation:
M =
=
Z
Z
Dn̂0 Dn̂01 · · · Dn̂0z−1 δ(n̂0 − n̂01 ) · · · δ(n̂0 − n̂0z−1 ) |n̂0 i hn̂01 | · · · n̂0z−1 Dn̂ |n̂i hn̂| · · · hn̂|
and,
T =
Z
Dn̂Dn̂0 T (n̂, n̂0 ) |n̂i hn̂0 | .
(6.8)
(6.9)
Thus, equation (6.5) becomes
0
ψ = M T ψ 1 ⊗ · · · ⊗ T ψ z−1 .
(6.10)
We now focus on the stability of the paramagnetic state against Néel ordering.
Hence, we have to use boundary conditions that are consistent with this kind of
ordering. As discussed at the end of Sec. 6.2, one can either use free boundary
spins with lower S, or connect the boundary spins to some fixed additional spins:
in the latter case, the additional spins must all have the same orientation to allow
the Néel ordering. This remark is particularly important because, as we will see
later in Sec. 6.5, on a random regular graph the boundary conditions on any given
tree-like subregion are fixed self-consistently, and in general are not consistent with
Néel ordering, leading to a disordered spin glass state. Assuming uniform boundary
conditions, we obtain the unique state at depth d − 1 by merging the z − 1 states at
level d using the T and M operators:
|d − 1i = M (T |di)⊗(z−1) .
(6.11)
with the Dirac inner product is not necessarily 1 for a properly normalized probability distribution.
Since the probabilistic L1 norm is incompatible with the Hilbert space structure, it is much simpler
to work with unnormalized vectors and keep in mind that a probabilistic interpretation only applies
in the standard basis.
124
The natural basis to work in is that of states with definite angular momentum,
i.e. states |l mi, which are eigenstates of the angular momentum operators L2 , Lz .
In the coordinate basis, these are simply the spherical harmonics, and as shown in
Appendix 6.A, they are eigenstates of the transfer matrix with eigenvalue λl .
It remains for us to understand exactly how the merge operation acts in the
angular momentum basis. If we insert resolutions of the identity in the angular
momentum basis into (6.8), we obtain
M =
=
Z
Z
Dn̂ |n̂i hn̂| · · · hn̂|
Dn̂
X X
l0 ,m0 l1 ,m1
···
X
lz−1 ,mz−1
|l0 , m0 i hl0 , m0 |n̂i
× (hn̂|l1 , m1 i hl1 , m1 | ⊗ · · · ⊗ hn̂|lz−1 , mz−1 i hlz−1 , mz−1 |)
X X
X
=
···
|l0 , m0 i (hl1 , m1 | ⊗ · · · ⊗ hlz−1 , mz−1 |)
l0 ,m0 l1 ,m1
×
Z
lz−1 ,mz−1
m
Dn̂ Yl0m0 ∗ (n̂)Yl1m1 (n̂) · · · Ylz−1z−1 (n̂)
(6.12)
For the case z = 3, the integral in (6.12) is simply the Clebsch-Gordan coefficient
that characterizes the fusion of two SU (2) spins. For higher values of z, this is the
appropriate generalization of the Clebsch-Gordan coefficient describing the fusion of
(z − 1) SU (2) spins. Thus we see that the merge operation, when written in the
angular momentum basis, has a natural interpretation as the fusion rules for the
O(3) symmetry group.
The paramagnetic state - here represented in Fourier space by the |00i state
- is always a fixed point: it is an eigenstate of the T -matrix, and the fusion of
any number of |00i states is again a |00i state. We proceed via linear stability
analysis: we introduce a perturbation into a state that is not uniformly weighted
on the sphere, and see if this grows or shrinks under the iteration procedure. We
note that we can decompose any such state into spherical harmonics, and so we write
125
|di = |00i + P∞
l6=0,m clm
|l mi into (6.11) to obtain
|d − 1i = M
=
λ0z−1
λ0 |00i + |00i
+(z − 1)λq−2
0
X
l6=0,m
X
l6=0,m
!⊗(z−1)
λl clm |l mi
λl clm |l mi + O(2 )
(6.13)
where we have used the fact that fusing any number of |00i states with an |l mi state
results in an |l mi state. We renormalize to leading order and find that the iterated
state is, to linear order
|d − 1i = |00i + (z − 1)
X λl
clm |l mi + O(2 )
λ
0
l6=0,m
(6.14)
The perturbation is irrelevant (shrinks under iteration) if the coefficient of the
linear term is less than 1, and relevant if it is greater than 1. The critical point is
reached when
λl
1
=
λ0
z−1
(6.15)
for any l. Using the temperature dependent expression (6.39) for the λl , one can show
that the dipole instability (l = 1) is the first one encountered as the temperature is
lowered, and therefore sets the transition temperature.
Using the results of Appendix 6.A (replacing β by the singlet parameter M ), we
obtain
Mc =
2
z−2
(6.16)
We see that for z = 2, 3, 4, Mc = ∞, 2, 1, while for all other values, Mc < 1. Since M
must be a positive integer, we see that for the chain (z = 2) all values of M correspond
to quantum-disordered states (which follows from the Mermin-Wagner theorem and
the original AKLT result [3]) whereas for z = 3, the M = 1 state is disordered while
126
the M = 2 state is critical, and finally for z = 4, the M = 1 state is critical. Bethe
lattices of higher connectivity will always have ordered AKLT ground states for any
value of M . See Fig. 6.1.
Finally, we consider the correlation function hn̂0 · n̂d i within the paramagnetic
phase. This is given by considering the response of hn̂0 i to a field on n̂d – which is the
same as asking how the dipole l = 1 perturbation propagates along a chain of length
d in a background of trivial l = 0 cavity states. This immediately implies
hn̂0 · n̂d i ∝
λ1
λ0
d
=
M
M +2
d
(6.17)
Notice that this implies that the naive correlation length never diverges – as usual with
tree models, phase transitions occur when the correlation decays at the same rate as
the growth of the lattice. For a slightly more detailed calculation, see Appendix 6.B.
As an aside, we note that we can use the same generalized transfer matrix technique to obtain the transition temperature for the Heisenberg model, a result first
obtained by Fisher [48] using a different method. This serves as a test of the technique
proposed here.
6.4
Variational bounds on the gap
We now perform a variational computation of the gap to excitations in the critical
model, similar to the Single-Mode Approximation (SMA) discussed in Ref. [12]. The
central idea of the SMA is to construct an excitation orthogonal to the ground state
by acting on it with a local operator, and then to reduce the energy of this excitation
by delocalizing it, thereby decreasing its kinetic energy. A variational bound on the
energy gap is given by
0 ≤ ∆ ≤ ∆SM A =
hΨSM A | H − E0 |ΨSM A i
hΨSM A |ΨSM A i
127
(6.18)
This approach is designed to optimize the energy due to the off-diagonal matrix
elements in the excited sector, which will be proportional to the usual graph Laplacian
for a nearest neighbor model. On the Bethe lattice, the spectrum of the Laplacian
is unusual. As argued in Ref. [79], there is necessarily a gap to hopping excitations
on tree-like graphs despite the existence of symmetry related ground states. Thus, in
some sense the SMA calculation is doomed to failure as it will never be able to close
this gap. Nonetheless, it is interesting to see how this plays out in an exact model.
We consider a rooted Cayley tree, and in order to restrict ourselves to studying
excitations confined to the bulk as the size of the tree grows we suppress excitations
far from the center using an infrared regulator λ. We therefore study variational
wavefunctions of the form
|ΨSMA i = |λi =
N
X
i=1
uλ (i)Siz |Ψi
(6.19)
where uλ (i) is a function only of the depth ρi of site i referenced to the root of the
Cayley tree. The SMA gap is
∆SM A
hλ| H − E0 |λi
= lim lim
λ→0 N →∞
hλ|λi
(6.20)
After some algebra, we may write (with the understanding that we always take N to
infinity before taking λ to zero):
∆SM A = lim
N →∞
λ→0
"P
z
z
i,j uλ (i)uλ (j)h[Si , [H, Sj ]]iΨ
P
z z
i,j uλ (i)uλ (j)hSi Sj iΨ
#
(6.21)
Asserting homogeneity of the graph, and noting that the Hamiltonian is a function
only of Si · Sj where i and j are nearest neighbors, we may re-express this as
∆SM A = lim
N →∞
λ→0
"
#
P
i,j uλ (i)Aij uλ (j)
f×P
|i−j| u (j)
λ
i,j uλ (i)γ
128
(6.22)
where f = 14 h[Siz − Sjz , [H, Siz − Sjz ]]iΨ , and we denote by A the adjacency matrix of
the Cayley tree; the correlations in the ground state are always exponential, and go
as γ |i−j| , where γ →
1
z−1
from below as we approach criticality. We have therefore
reduced the problem of the SMA on the Bethe lattice to understanding (i) the spectrum of the graph Laplacian (the adjacency matrix up to a sign) and (ii) the behavior
of the ground-state correlations.
2
Our choice for a variational ansatz is to take uλ (i) = e−λρi . This is motivated by
the fact that the number of sites at a given distance from the center grows exponentially, and therefore in order to remain in the bulk of the tree, we need to cut off the
wavefunction faster than exponentially.3 We perform the summations by converting
the sum over sites into a sum over depths, approximating the sums by integrals and
using steepest-descent. We find that, at criticality, the gap is nonvanishing:
∆G
SM A
"
#
√
z−2 z−1
f
√
∼ z
8 z − 1 + 4 z−1
log (z−1)
(6.23)
Excitations constrained to live in the bulk are therefore always gapped, even at crit√
icality. The factor of z − 2 z − 1 is precisely the spectral gap for bulk excitations
on a Cayley tree [79]. A state where uλ (i) is independent of position must be gapless
in the broken-symmetry phase of the model, since it connects the different brokensymmetry ground states. We cannot recover this state through a correctly regulated
calculation in the chosen order of limits, however.
There is a straightforward physical argument for this gap. By our choice of vari√
ational ansatz, we cut off the excitation at some depth D ∼ 1/ λ. On a Euclidean
lattice of dimension d, this costs a surface energy ∝ Dd−1 which is normalized by the
weight of the wavefunction ∝ Dd . As D → ∞ (λ → 0), the SMA gap therefore van3
In principle, an exponential regulator with a sufficiently fast decay constant also works. The
calculation proceeds in a similar fashion but the interpretation is more complicated and no more
enlightening.
129
ishes as D−1 . On tree-like lattices, both the surface area and the bulk normalization
scale as (z − 1)D ; the boundary is always a finite fraction of the bulk. Thus, the ratio
remains finite as D → ∞ and the gap cannot close [63, 79].
6.5
AKLT model on regular random graphs: frustration and the spin glass state
We now consider the same model on a regular random graph [64] of connectivity z.
The ensemble of these graphs is constructed by assigning uniform probability to all
possible graphs of N vertices, such that each vertex is connected to exactly z links.
We remind the reader that the thermodynamic limit of this ensemble reproduces
locally tree-like infinite graphs with no boundary (see Sec. 4.3). Typical graphs are
characterized by many large loops of even and odd length; this strongly frustrates
antiferromagnetic ordering, which gives way to a spin glass phase instead4 . This last
observation is particularly interesting when carried over to the quantum problem.
The reasoning outlined in section 6.2 clearly applies to the random graph model,
whose AKLT ground state is therefore described by a classical Hamiltonian of the
form (6.3), where the pairs hi, ji are connected by a link of the random graph. The
main difference between the tree model and the random graph model is that the
recurrence equation (6.5) now does not hold for the full graph; it only holds for a
tree-like subregion of the graph, and has to be initialized using the boundary values
of the ψ i (n̂i ) that are determined by the summation over the rest of the graph. In
other words, the recurrence on the subregion is initialized from random self-consistent
boundary conditions, determined by the rest of the system: these boundary conditions
are not consistent with Néel ordering, which is therefore frustrated, as discussed in
4
See Ref. [88] for a general discussion and Ref. [74] for the explicit computation of the phase
diagram of a classical Ising antiferromagnet.
130
Sec. 6.3 above. However, since the tree-like subregions grow in size when N → ∞,
equation (6.5) must be iterated a very large number of times. One can classify the
different phases of the system by studying its fixed points [88, 87].
To calculate the stability of the paramagnetic solution against spin glass ordering
we observe that the spin glass transition is signaled, as usual, by the divergence of
the classical spin glass susceptibility [15]:
χSG =
1 X
[hn̂i · n̂j i]2 .
N ij
(6.24)
The details are discussed in Appendix 6.B; the result is that χSG is finite if for all l
1
λl
≤√
.
λ0
z−1
(6.25)
Once again the instability originates in the l = 1 sector and occurs at
MSG = √
2
z−1−1
(6.26)
We see that MSG = ∞ for z = 2, MSG = 4.828 for z = 3 (hence the system is a spin
glass for M ≥ 5), MSG = 2 for z = 5 and MSG = 1 for z = 10 (in these cases the
system is critical), and it is smaller than 1 for any z > 10. See Fig. 6.1.
With the Néel-ordered phase suppressed, the quantum paramagnet extends further in the z − M plane than on the Cayley tree models. Nonetheless, it is clear that
the paramagnetic solution develops an instability to spin glass ordering at large M .
What are the properties of this low temperature phase? Most of the detailed work on
classical spin glasses has focussed on discrete models. The AKLT mapping provides
a classical vector model with weakly divergent interactions whose glass phase has
not yet been studied. Nonetheless, most of the qualitative features of the classical
multiple-valley picture should still hold and these provide an intriguing scenario for
131
Figure 6.1: Phase diagrams for AKLT model with singlet parameter M on tree-like
lattices with coordination z. On the Cayley tree the transition is from paramagnetic
(PM) to Néel-ordered (AF) phase at the solid blue line with no spin glass. On regular
random graphs the transition is from the paramagnetic to spin glass ordered (SG)
phase at the dashed green line — there is no antiferromagnet. The models with Bethe
lattice critical correlations are labeled with large dots.
the quantum system. The classical Gibbs measure decomposes into a collection of
clustering pure states α = 1 . . . N with essentially disjoint support. In each of these
states, the ψαi (n̂i ) – and thus the local magnetizations – are macroscopically different.
This strongly suggests that the quantum AKLT ground state |Ψi itself is a superposition over a collection of macroscopically distinct degenerate ground states |Ψα i
each of which corresponds to one of the classical clustering states.
While we believe that the above picture holds in general, a rigorous derivation is
problematic. In the following we will attempt to justify it in more detail and point
out some of the subtleties that must be dealt with. First, there does not yet exist
a detailed study of the classical vector model5 – for the purposes of this chapter, we
shall assume that, modulo the O(3) global symmetry, the qualitative behavior is that
of the better studied Ising antiferromagnet on a regular random graph [74].
By analogy to this model, two different phases exist: at high temperatures, the
stable phase is a paramagnet where ψ i (n̂i ) = ψ(n̂) = 1 is the same for all sites, and
5
Although there is a replica symmetric treatment of a related model with the additional complication of random interactions. See Ref. [32].
132
it is the unique fixed point of Eq. (6.5). At low temperatures, the stable phase is a
spin glass, characterized by the existence of many non-symmetry related pure states
labeled by α = 1, . . . , N within which connected spatial correlators vanish. This is
related to the existence of many different fixed points of Eq. (6.5), and reflects the
decomposition of the thermodynamic Gibbs measure P = exp(−βHcl )/Z as follows:
P ({n̂}) =
N
X
α=1
wα
Z
dg Pα ({g · n̂})
(6.27)
where the Pα are representative classical pure state measures. By integrating over
g, the O(3) of global rotations of spin space, we account for the continuous family
of symmetry related pure states associated to each representative state α. One can
access these representative pure states by adding a uniform infinitesimal field to the
classical model, but as the quantum AKLT state is a singlet, we prefer to work without
explicitly breaking this symmetry [15].
The Ising spin glass models with two-body interactions which have been studied,
such as the Sherrington-Kirkpatrick model [90] and the random graph antiferromagnet [74], are characterized by a continuous spin glass transition6 with a finite collection
of pure states throughout the spin glass phase7 . We shall therefore assume that this
is true of our collection of representative pure states. Indeed, all we will need is that
N grows at most polynomially in N as the thermodynamic limit is taken.
The pure state decomposition (6.27) has striking consequences for the structure
of the low-energy states of the quantum AKLT Hamiltonian. To wit, we argue that
Z
N
X
√
wα
dg g |Ψα i
|Ψi =
(6.28)
α=1
6
Other mean field spin glass models, such as the p-spin model, can have a number of states
scaling exponentially in system size. These models show a discontinuous spin glass transition. However, antiferromagnetic models with two-body interactions usually do not show this phenomenology,
therefore we will not investigate this transition in this chapter.
7
See Ref. [90], and in particular the reprint on page 226, for a more detailed discussion of this
delicate statement.
133
up to exponentially small corrections in the thermodynamic limit, where the |Ψα i
can be interpreted as a collection of symmetry breaking degenerate quantum ground
states whose correlations correspond to the classical pure states Pα .
We argue this in three parts. First, we assume the existence of a collection of
quantum states |Ψα i such that
|h{n̂i }|Ψα i|2 = Pα ({n̂i })
(6.29)
and show that the quantum state Eq. (6.28) reproduces the observables of the classical
decomposition of Eq. (6.27). Second, we will show that up to exponentially small
corrections each of the |Ψα i are themselves orthogonal ground states. Finally, we
address the issue of the existence of such states given the classical distributions Pα .
The first part follows the argument of Ref. [16] but we rephrase it in terms of
density matrices. Consider the density matrix of the proposed state Eq. (6.28):
ρ = |ΨihΨ| =
X√
wα wβ
α,β
Z
dg
0
Z
dg g 0 |Ψα ihΨβ |g †
(6.30)
Given any local operator Ô depending only on spins8 , its expectation value in state
|Ψi is given by
hÔi = Tr ρÔ
Z
Z
h
i
X√
0
wα wβ dg
dg Tr g 0 |Ψα ihΨβ |g † Ô
=
α,β
=
X√
α,β
wα wβ
Z
dg
0
Z
8
dg hΨβ |g † Ôg 0 |Ψα i
(6.31)
Such observables are diagonal in the coherent state basis and therefore their correlations follow
from the classical measure.
134
We now argue that the interference term is negligible. That is,
hΨβ |g † Ôg 0 |Ψα i = δαβ δg g0 hΨα |g † Ôg|Ψα i
(6.32)
up to exponentially small corrections in the thermodynamic limit. This follows from
the observation that |Ψα i and |Ψβ i have macroscopically distinct magnetization patterns that completely break the O(3) symmetry. In particular, the classical configurations {n̂i } on which the wavefunction |Ψα i is concentrated have extremely small
weight in any other wavefunction β 6= α – this remains true even with arbitrary global
rotations allowed between them. If α = β but g and g 0 differ, then the configurations
with weight are again macroscopically distinct by virtue of the net global rotation
g −1 g 0 between them. The fact that the observables are local and have bounded matrix
elements does not modify these assertions. Since N is finite for our antiferromagnetic
model, the finite sum over the exponentially small corrections does not modify the
result:
hÔi =
X
wα
α
Z
dg hΨα |g † Ôg|Ψα i
(6.33)
Inserting a complete set of coherent states and using Eq. (6.29) reproduces the classical distribution Eq. (6.27). The probability of finding the quantum system in a state
α (with respect to the state |Ψi) is the same as that of the classical problem: both
are given by wα .
Furthermore, the |Ψα i must have exponentially small energy with respect to the
quantum Hamiltonian. Since hHi = 0 in the AKLT state, using equation (6.33) and
the rotational invariance of H, we have that
0 = hHi =
X
α
wα hΨα |H|Ψα i
135
(6.34)
up to exponentially small corrections. Since each of the terms in the sum is nonnegative, it follows that
hΨα |H|Ψα i . O(e−N )
(6.35)
Thus, the |Ψα i are a collection of nearly orthogonal, nearly zero energy states, each
of which generates a further continuous collection of such degenerate states under the
action of O(3).
Finally, we turn to the slightly thorny question of whether states satisfying
Eq. (6.29) exist.
The problem is that the coherent state basis is overcomplete
for any fixed spin size S = zM/2 and we cannot necessarily find quantum states
which have given expansions in this basis. That is, a priori we cannot simply set
p
h{n̂i }|Ψα i = Pα ({n̂i }) and know we have a well-defined quantum state for spins of
size S. In the large spin limit, there is no problem as the coherent states become a
complete, rather than overcomplete, basis. This coincides with the zero temperature
limit of the classical companion model and thus the pure states |Ψα i may be identified
with the (many degenerate) minima of the energy function (6.3). At finite M , we
cannot find such finely localized states in the coherent state representation – the
most localized state has solid angular scale ∼ 1/M – but the finite temperature
fluctuations around the classical minima will also smear the Pα at a similar scale.
On the other hand, we already used above that the states have disjoint support, up
to exponentially small corrections in N . For a given classical configuration {n̂i }, only
one state contributes to h{n̂i }|Ψi significantly. Thus, in a given region of classical
configuration space, h{n̂i }|Ψi coincides with one of the h{n̂i }|Ψα i, and conversely,
each of the h{n̂i }|Ψα i can be seen as the restriction of the full h{n̂i }|Ψi to that
region. Since the 1/M smoothing is local in configuration space, it is safe to assume
that the states h{n̂i }|Ψα i are as smooth as the original h{n̂i }|Ψi, therefore |Ψα i can
be defined without ambiguity. Based on the above arguments, we think it likely that
136
at least an approximate pure state decomposition of the form proposed above can be
found even at finite M .
In summary, we obtain the following picture for the low energy spectrum in the
spin glass phase: the non-clustering paramagnetic AKLT ground state |Ψi can be
decomposed in a superposition of several almost degenerate states, whose energies are
of order exp(−N ). These states enjoy the clustering property (vanishing of connected
correlations) and are characterized by amorphous order (the local magnetizations are
different in each state). It would be nice to check this scenario explicitly by means of
exact diagonalization.
It would be very interesting to obtain more detailed information on the spectrum
of such spin glass Hamiltonians. For instance, a natural question is whether there
is an energy gap between the degenerate low-lying spin glass states and the excited
states. Indeed, we expect Goldstone (or Halperin-Saslow) modes [58] associated with
twisting of the amorphously magnetized states |Ψα i. While it is difficult to explicitly
construct a coarse-graining procedure to produce an effective theory of such modes
on a tree-like graph, one usually expects that such a theory applies in sufficiently high
dimensions [57]. Insofar as the effective theory is an elastic hydrodynamics living on
a tree-like graph, the corresponding modes should remain gapped, as discussed in
Chapter 8. This suggests that the low energy spectrum is indeed gapped in any given
pure state sector.
6.6
Concluding remarks
In this chapter, we extend the study of AKLT models to locally tree-like graphs
of fixed connectivity by exploiting the quantum-classical mapping of the associated
wavefunctions. On the infinite Cayley tree, we recover the results obtained in Ref. [42].
We find that the Bethe lattice possesses the peculiar property that it is possible to
137
choose parameters (for z = 3, 4) so that the corresponding AKLT state is critical. A
variational calculation of the gap is unable to produce gaplessness, which is consistent
with the arguments of Chapter 8 that this is a general feature of locally tree-like
graphs: essentially, one cannot deform a uniform excitation into long-wavelength
rotations of the order parameter, without jumping a gap in the Laplacian spectrum.
Turning to regular random graphs, we find that the companion classical model
is unstable to spin glass ordering within a cavity analysis. This is a general feature
of classical antiferromagnetic models on such graphs, but has striking consequences
given that the peculiarities of such mean-field-like glasses should directly transfer to
the quantum ground state of the AKLT model. This provides an alternative route to
the study of quantum glassy order in tree-like models (see for example Ref. [81, 71, 70,
114, 26, 66]). We argue that there are now many (nearly) degenerate quantum ground
states with macroscopically distinct magnetization patterns, but that there remains
a gap to Halperin-Saslow waves for geometric reasons analogous to the simpler case
of the antiferromagnet.
There are several avenues for future research. One obvious direction is to study
the classical vector spin glass and the corresponding classical measure. In a different
vein, we observe that the AKLT construction applies at a very special point in the
space of quantum Hamiltonians. To what extent do the features of the quantum
AKLT glass extend to regions proximate to this exactly solvable point? Ideally, the
AKLT glass would capture the essential features of a broader range of quantum spin
glasses, playing a role reminiscent of that played by the S = 1 AKLT chain in relation
to the Haldane phase.
138
6.A
Transfer matrix for the AKLT model
The transfer matrix may be thought of as a map between functions defined on the
sphere:
Z
(T f )(n̂) =
dn̂0
T (n̂, n̂0 )f (n̂0 )
4π
(6.36)
where n̂, n̂0 ∈ S N −1 . In our case, as in most cases of interest, the transfer matrix is
a rotational scalar and we can work in the angular momentum basis. The eigenvalue
must depend only on the L2 eigenvalue l and not on the Lz eigenvalue m. It therefore
suffices to solve the problem in the case m = 0.
We wish to solve the eigenvalue equation:
λl fl (n̂) =
Z
S2
Dn̂0
T (n̂, n̂0 )fl (n̂0 )
4π
(6.37)
Since the kernel T depends only on n̂ · n̂0 = cos θ, we work in polar coordinates with
the z-axis along n̂ and substitute x = cos θ to obtain
1
λl fl (1) =
2
Z
1
dx T (x)fl (x)
(6.38)
−1
A natural guess for the eigenfunctions is that they are Legendre polynomials. The
transfer matrix in our case is T (x) = ( 1+x
)β .Using standard identities,
2
λl,AKLT
[Γ(β + 1)]2
=
Γ(β − l + 1)Γ(β + l + 2)
(6.39)
A similar discussion for the Heisenberg model for arbitrary N and the case of SU (N )
and Sp(N ) groups, may be found in Ref. [44].
139
6.B
Stability against spin glass ordering on a regular random graph
We follow closely the derivation of Ref. [124], Appendix A. In the thermodynamic
limit, the spin glass susceptibility
χSG =
1 X
[hn̂i · n̂j i]2
N ij
(6.40)
can be rewritten, by taking the average over the random graphs and using translational invariance, as
χSG =
∞
X
d=0
Nd [hn̂0 · n̂d i]2
(6.41)
where Nd is the number of sites at distance d from a reference site. The sum is
convergent as long as
lim (Nd )1/d [hn̂0 · n̂d i]2/d ≤ 1
d→∞
(6.42)
Note that (Nd )1/d → z − 1 for large d. In the paramagnetic phase, hn̂0 · n̂d i is given
by the response of hn̂0 i (the root) to a small magnetic field coupled to n̂d , a leaf at
distance d, of a tree whose other nodes are in the paramagnetic state |00i. Hence we
get (repeated indices summed):
dhn̂i0 i
=
hn̂0 · n̂d i ∝
dhid
Z
dn̂0 n̂i0
dψ 0 (n̂0 )
dhid
(6.43)
Clearly the term that gives the exponential dependence on d is the variation of ψ 0 (n̂0 )
with respect to hd . Using the recursion relation (6.5) we can rewrite it as
dψ 0 (n̂0 )
=
dhd
Z
dn̂1 · · · dn̂d
dψ 0 (n̂0 ) dψ 1 (n̂1 )
dψ d (n̂d )
·
·
·
dψ 1 (n̂1 ) dψ 2 (n̂2 )
dhd
140
and the exponential dependence is related to the eigenvalue of the transfer matrix
dψ d (n̂d )
dψ d+1 (n̂d+1 )
= T (n̂d , n̂d+1 ). These can be obtained by repeating the analysis
of section 6.3. Indeed if we use the same ket notation and write the variation
P
δ |d + 1i = ∞
l6=0,m clm |l mi we can use the results of section 6.3 to obtain
δ |di = X λl
clm |l mi + O(2 )
λ0
l6=0,m
(6.44)
The absence of the factor (z − 1) with respect to Eq. (6.14) is due to the fact that
here we are only varying one of the neighbors of a given spin, the neighbor on the
path linking the root to the given leaf at distance d. The relevant eigenvalues are
therefore λl /λ0 and we obtain the condition
(z − 1) max{(λl /λ0 )2 } ≤ 1
for the convergence of χSG .
141
(6.45)
Chapter 7
Griffiths-McCoy singularities and
Lee-Yang zeros in a solvable
diluted ferromagnet
7.1
Introduction
In the present chapter,1 we study the diluted Ising ferromagnet on the Bethe lattice
as a case study for the application of the cavity method to problems with GriffithsMcCoy (GM) singularities [56, 119, 84] in their thermodynamics in an applied field.
While the form of these singularities can be readily determined from rough estimates
of the statistics of the rare regions from which they emanate, their detailed extraction
can be a tricky task due to their delicate nature, especially in classical systems2 .
Indeed, in field theoretic formulations they appear as non-perturbative (instanton)
effects [36, 37].
1
Chapter based on work with A. Scardicchio and S. L. Sondhi [82].
For example, see the numerical work on the two dimensional version of the problem studied in
[108].
2
142
Here we consider the task of extracting these GM singularities from the cavity
method. The method is exact on trees and hence the functional recursion relation
to which it gives rise must contain the GM physics which is known to exist. The
challenge is to extract it by constructing the appropriate fixed point solution. We
study the particular case of a diluted ferromagnet on the Bethe lattice which has
an extended GM region at low temperatures and large dilution. While the general,
exact, determination of GM singularities everywhere in the phase diagram is a hard
problem, we are able to solve the problem in the infinite coupling limit made precise
below. Here we can directly solve the cavity equations in a field and relate the
solution to the statistics of clusters and to the density of Lee-Yang zeroes commonly
used to characterize GM effects. Further, in this limit the phase transition between
the paramagnet and the ferromagnet is itself essentially of a GM character and its
critical behavior, which we extract, can be viewed as an enhanced GM phenomenon.
The problem of the dilute Bethe lattice ferromagnet and of GM singularities has
been considered before [24, 14] by different methods. As our interest is primarily
in the development of the cavity method, we give a self-contained presentation from
that viewpoint.
7.2
Model and organization
We consider the following disordered Ising Hamiltonian on the Bethe lattice with
connectivity q:
βH = −J
where
X
hiji
ij σi σj − H
X
σi ,
i


 1 with probability p;
ij =

 0 with probability 1 − p.
143
(7.1)
The random couplings ij indicate the presence or absence of a bond in the diluted
Bethe lattice. For probability p < pc = 1/(q − 1) the lattice has no giant clusters and
the density of large finite clusters decays exponentially. For p > pc , giant clusters exist
with finite density and at the percolation transition, p = pc , the density of clusters of
size n develops a long algebraic tail Wn ∼ n−5/2 (independent of q) [49]. We note that
the dimensionless coupling constants J and H differ from the conventional magnetic
exchange and field by factors of inverse temperature β = 1/T . The limit T → 0 with
J H will be denoted as the J = ∞ limit.
In Section 7.3, we provide a guided tour of the well-known phase diagram of this
model from the point of view of the cavity method. We then establish the critical
behavior at the phase transitions using a set of recursion relations for the moments of
the cavity field distribution. We also show that the critical behavior can be extracted
via a simple numerical algorithm, which we discuss in some detail in the final Section
7.8.
The following sections are devoted to investigating exact analytic results in the
infinite (dimensionless) spin-spin coupling limit, J = ∞. This corresponds to the
horizontal axis of the phase diagram in Fig. 7.1. In Section 7.4, we find an explicit
expression for the magnetization M (H) by means of a sum over connected clusters,
which follows the standard GM treatment due to [59]. In Section 7.5 we show that
the same expression can, in fact, be extracted from the cavity method. In this limit,
the magnetization goes to zero with the field for p ≤ pc = 1/(q − 1) while for p > pc a
spontaneous magnetization develops. We first show that: 1) for p < pc the asymptotic
series expansion for the magnetization contains only integer powers M (H) = χH +
c3 H 3 + ... and 2) on the contrary at p = pc the series expansion contains semi-integer
√
powers as well M (H) = c1/2 H + c1 H + c3/2 H 3/2 + .... That is, the critical exponent
δ = 2 at p = pc , J = ∞.
144
Figure 7.1: Phase diagram for the diluted ferromagnet on the connectivity q = 3
Bethe lattice. The model is paramagnetic (PM) for any coupling weaker than JG . It is
ferromagnetic above the percolation transition (pc = 1/2) for strong enough couplings
Jc (p) ≥ JG and there are essential singularities but no spontaneous magnetization
throughout the Griffiths-McCoy region (GM).
In Section 7.6 we develop an alternative integral representation for M (H) that
corresponds to a harmonic expansion. This representation will allow us to calculate
the (smoothed) density of Lee-Yang (LY) zeros ρsm at J = ∞ on the imaginary H
axis (θ = Im H) in Section 7.7 and to show that for p < pc a GM phenomenon indeed
occurs, i.e. the density of zeros is non-zero and vanishes as e−α/θ when approaching
the origin. For p = pc we find α = 0 and the density vanishes as the power law
√
ρ ∝ θ.
7.3
Phase diagram and cavity equations
The p—J phase diagram (Fig. 7.1) of the diluted ferromagnet H is physically well
understood and can be derived naturally in a cavity method formalism [120, 88]. In
this approach, one considers the flow of cavity fields from the boundaries of the tree
inward toward the center. A cavity field hi on a spin σi at a distance d from the
boundary describes the spin’s magnetization in the absence of the link connecting it
145
to the next spin inward. The cavity field hi only depends on the cavity fields on σi ’s
neighbors at distance d − 1 and therefore one can define a natural flow for the depth
dependent distribution of fields P (d) (h):
P (d) (h) = E
Z
q−1
Y
i=1
!
dhi P (d−1) (hi ) δ h −
q−1
X
!
u(hi + H, Ji )
i=1
(7.2)
where
u(hi + H, Ji ) = tanh−1 (tanh(Ji ) tanh(hi + H))
(7.3)
gives the bias on the field h due to a spin σi connected through a link Ji . E is the
expectation with respect to the ij distribution. Fixed point distributions P (∞) (h)
describe the statistical features of the bulk (central region) of the Bethe lattice. In
order to break the Ising symmetry, we will always assume an infinitesimal uniform
positive boundary field P (0) (h) = δ(h − 0+ ) as the starting point for the flow.
In the undiluted model, p = 1, all of this discussion reduces to the simple BethePeierls mean field theory for a connectivity q lattice. Since there is no randomness,
the cavity field distributions P (d) are simply delta functions located at, possibly depth
dependent, fields h(d) . Equation (7.2) reduces to a flow equation for h(d) :
(d)
h
=
q−1
X
u(h(d−1) + H, J)
i=1
= (q − 1) tanh−1 tanh(J) tanh(h(d−1) + H)
For J < JG = tanh−1
1
q−1
(7.4)
, the flow at H = 0 has only one fixed point h(∞) = 0 cor-
responding to the paramagnetic phase. For J > JG , the h(∞) = 0 fixed point becomes
unstable to a spontaneously magnetized ferromagnetic fixed point with h(∞) > 0.
Expansion of the fixed point equation to leading order in H and = (J − JG ) gives
146
the well-known mean-field critical exponents at J = JG :
M (H, J = JG ) ∼ H 1/δ ;
δ
β
M (H = 0, J > JG ) ∼ (J − JG ) ;
= 3
(7.5)
β = 1/2
Under dilution, we must return to the more general cavity distribution flow defined
by equation (7.2) to extract the phase behavior. Notice that the paramagnetic cavity
distribution P P M (h) = δ(h) is always a fixed point of the flow at H = 0, just like
h(d) = 0 is always a solution for the undiluted model. As in the undiluted case, this
fixed point will become unstable above some critical coupling Jc (p). Near P P M (h)
(i.e. for small h), we consider the linear stability of the first moment of P (d) (h):
hhi(d) = E
= E
Z
Z
dh
q−1
Y
i=1
q−1
Y
i=1
!
dhi P (d−1) (hi ) δ h −
q−1
X
i=1
! q−1
X
u(hi , Ji )
dhi P (d−1) (hi )
≈ (q − 1) (E tanh(J)) hhi
ui
!
h
i=1
(d−1)
to leading order. Thus, 1 = (q −1)p tanh(Jc (p)) gives the critical boundary separating
a stable paramagnetic phase from the ferromagnetic phase. A small rearrangement
gives:
Jc (p) = tanh
−1
pc
p
(7.6)
This agrees precisely with the undiluted critical point Jc (p = 1) = JG found above
and also predicts that for p < pc the paramagnetic phase persists for all finite J.
There is no ferromagnetic phase transition for a model with only finite clusters, as
one expects.
In order to extract the critical behavior along the diluted phase boundary, we
wish to expand the fixed point equations near the critical solution as we did in the
discussion of the undiluted model. Rather than working with equation (7.2) directly, it
147
is more natural to use an equivalent infinite set of recursion relations for the moments
of P (h). These can be derived by multiplying both sides of equation (7.2) by hn and
integrating or by considering the relation on random variables
0(d)
h
=
q−1
X
i=1
(d−1)
)
i tanh−1 tanh(J) tanh(hi
taken to the power n and averaged. Near P P M (h), we expand this relation around
small hi :
0
h =
q−1
X
i=1
1
i τ (hi − (1 − τ 2 )h3i ) + ...
3
(7.7)
where τ = tanh J and we have suppressed the depth superscripts.
Sufficiently near the phase boundary, we expect the moments hhn i to decrease
exponentially with n and thus only a few leading order moments need be retained to
extract the leading critical behavior at finite J. Taking powers of equation (7.7) and
averaging, we find
2
hh0 i = 2pτ hhi − pτ (1 − τ 2 ) h3
3
02 2
2
h
= 2pτ h + 2p2 τ 2 hhi2
03 h
= 2pτ 3 h3 + 6p2 τ 3 hhi h2
to cubic order. We have specialized to the case q = 3 in order to simplify the
presentation; for q > 3 an additional term at cubic order is generated but the critical
exponents remain the same.
Near the phase boundary in the p—J plane, we can define a small parameter by
writing 2pτ =
p
pc
tanh J = 1 + . We will treat the fixed point equations to leading
order in the expansion. For < 0 the only real solution is paramagnetic:
hhi = h2 = h3 = 0.
148
(7.8)
Figure 7.2: Lyapunov exponents of the ferromagnetic fixed point of the iteration
equations for J = 1. Notice that for < 0.13... they are all negative, signaling
stability of the solution. At larger , the stationary point becomes a focus before
eventually becoming unstable.
This solution is stable since its local Lyapunov exponents (, − ln τ1 , − 2 ln τ1 ) are
all negative.
For > 0 this solution becomes unstable. We find two other ferromagnetically
ordered solutions, which are linked by the symmetry h → −h, and choose the positive
one. This is
hhi = 2
r
1 − τ 1/2
+ ...
τ
2
2
h
=
+ ...
τ
3
6
3/2
h
= p
+ ... .
τ (1 − τ ) 1 + τ
(7.9)
One can find the Lyapunov exponents of this stationary point analytically but the
expressions are unenlightening. We plot a typical case in Fig. 7.2. Notice that the
results (7.9) are consistent with the assumption that hhn i decreases exponentially
with n.
From Eq. (7.9) one can read off the critical exponent β = 1/2 as the power of in hhi ∼ m. This is valid for all τ < 1 and sufficiently small . The point τ = 1
149
(J = ∞) is different and needs to be treated more carefully. As τ → 1 the coefficient
of 1/2 in (7.9) vanishes, which implies that the J = ∞ critical exponent β 0 > 1/2
while the divergence of the coefficient of 3/2 means that β 0 < 3/2. Indeed, from the
exact solution of Sec. 7.4, we will find β 0 = 1.
For sufficiently large > c , the Lyapunov exponents become positive, signaling
a loss of stability of the third order ferromagnetic solution (for the value J = 1, τ =
tanh(1) in Fig. 7.2, c = 0.137... ). This indicates that the first few moments flow
to large scale and our truncation to cubic order fails. The value of c decreases
monotonically as τ approaches 1 and to accurately find the fixed points we need
to keep track of more moments of h in our iteration equations. At this point it
is convenient to switch to numerical solution of the full cavity equation (7.2) by
population dynamics, as described in Sec. 7.8.
Having explored both above and below the critical point, we return briefly to
the critical point at = 0. Here, at linear order, there is a marginal flow near the
paramagnetic fixed point. It is possible to analyze the truncated flow equations (7.8)
at higher order to discover that the paramagnetic solution is indeed algebraically
(rather than exponentially) stable, as one expects of a second order phase transition.
With some additional algebra it is possible to carry a small applied field H through
all of the above arguments at = 0 and show that the critical exponent δ = 3 all
along the p > pc phase boundary.
The final important feature of the phase diagram is the presence of GM singularities throughout the p < 1, J > JG region. That is, the density of LY zeros on the
0
imaginary H axis of the partition function has an essential singularity like e−a /Im H
throughout this region due to the cumulative influence of rare large undiluted regions
and there is therefore no gap. Equivalently, the real magnetization M ∼ e−a/H in a
real applied field H. Although this can be seen from elementary rigorous arguments
[56, 119], it is difficult to detect either analytically or numerically at finite J. However
150
in Sections 7.6 and 7.7 we will use the exact solution of the cavity equations at J = ∞
to exhibit these essential singularities explicitly and subject them to detailed study.
7.4
Cluster series at J = ∞
For the remainder of the chapter, we will focus primarily on the J = ∞ part of
the phase diagram of the model. We first review the classic argument due to Harris
[59] based on an expansion over connected clusters. This will lead to an exact series
expansion for M (H) that we will independently rederive using the cavity approach
in Sec. 7.5.
Consider a cluster of n + 1 spins connected by n bonds. For J H ∼ 1 (which
is the meaning of the J = ∞ limit) each connected cluster behaves like a piece of
ferromagnet. Indeed for H = 0 there are two degenerate ground states, one with
all spins pointing up and one with all spins pointing down. The first excited states
are spin flips at energy ∼ J above the ground states and their presence is negligible.
Turning on a magnetic field H the degeneracy is broken and (if H is positive, say)
the state with all spins pointing up is energetically preferred. Therefore the cluster
will acquire a small magnetization:
Mn (H) = (n + 1) tanh((n + 1)H).
(7.10)
The total magnetization per spin is obtained by summing over all the clusters with
their weights Wn , corresponding to the number of clusters of size n per spin3
M (H) =
X
Wn Mn (H).
(7.11)
n≥0
3
Notice that in this approach the infinite coupling limit is solvable because the magnetization of
a cluster depends only on the number of spins in the cluster and not on its detailed shape.
151
This equation has been studied before and results can be found in [59] for the magnetization, and [41] for the scaling law of the magnetization at the critical point.
We will reproduce those results on the magnetization for completeness, but the main
focus of this chapter will be the density of Lee-Yang zeros and the solution of cavity
field equations from which we will recover the known results.
From the solution of the bond percolation problem on the Bethe lattice [49] the
number per spin Wn of clusters of bond size n is given by
Wn (p) = q
((n + 1)(q − 1))!
pn (1 − p)n(q−2)+q .
(n + 1)!(n(q − 2) + q)!
For simplicity, we consider q = 3 in the following as all of the essential physics are
already present. We can easily obtain the asymptotic behavior of the magnetization
by using the asymptotics of Wn as
12
Wn ∼ √ (1 − p)3
π
5/2
1
e−nA(p)
n
(7.12)
where
A(p) = ln
1
.
4p(1 − p)
(7.13)
A(p) is the exponent governing the decay rate of the cluster sizes and it will appear
often in the remainder of the chapter. For p < pc = 1/2, A > 0 and Wn decreases
exponentially. For p = pc , A = 0 and we have instead a power-law decay with
exponent 5/2 (the exponent is independent of q):
3
Wn ∼ √
2 π
5/2
1
.
n
(7.14)
This change in the asymptotic fall-off of the cluster distribution at criticality is the
reason for the change in the response to an applied external field at zero temperature.
152
Indeed we can easily see how this works. For A > 0 and small H we can write an
asymptotic expansion by expanding the tanh in the sum (7.11):
M (H) =
X
Wn (n + 1) tanh((n + 1)H)
n
' H (n + 1)2 + O H 3
1+p
= H
+ O H3 ,
1 − 2p
(7.15)
which is linear in H. At the percolation threshold, however, A = 0 and hn2 i diverges.
The expansion of tanh inside the first sum is unjustified. To find the first term in the
asymptotic expansion of M (that we will derive in a formally correct way in Sec. 7.6)
we use instead (7.12):
X 3 1 3/2
√
M (H) '
tanh(nH)
n
2
π
n
Z ∞
3
3 √
' √
H
dx x− 2 tanh x + O (H) .
2 π
0
(7.16)
So the susceptibility diverges although there is no spontaneous magnetization [59].
We now turn to a derivation of the above results from the cavity method.
7.5
Cavity approach at J = ∞
The cavity method for this system gives a probability distribution for the cavity fields
which satisfies the fixed point equation (cf. Equation (7.2))
P (h) = E
Z Y
q−1
i=1
dhi P (hi )δ h −
q−1
X
!
u(hi + H, Ji )
i=1
(7.17)
In the J → ∞ limit, we can linearize the cavity biases u:
u(h + H, Ji ) = (h + H)i
153
(7.18)
At q = 3, the fixed-point equation (7.17) becomes
P (h) = (1 − p)2 δ(h) + 2p(1 − p)P (h − H)
Z
2
+ p
dh2 P (h2 − 2H)P (h − h2 ).
This equation can be solved by defining the Laplace transform
g(s) =
Z
∞
dhP (h)e−sh ,
0−
making sure to include the delta function at h = 0. The equation for g is quadratic
0 = p2 e−2sH g(s)2 + (2p(1 − p)e−sH − 1)g(s)
+ (1 − p)2 .
(7.19)
with solution
3Hs
e2Hs − 2eHs (p − p2 ) − e 2
g(s) =
(2p2 )
p
eHs − 4(p − p2 )
.
(7.20)
The second solution to (7.19) is not physical. Even without inverting the Laplace
transform all the properties of the solution can be extracted from g(s). For example
the normalization condition, the zeroth moment, is


 1
∞
if p ≤ 1/2;
dhP (h) = g(0) =
2

0
 (1−p)
if p > 1/2.
p2
Z
and the first moment is
hhi = −
∂g 2Hp
=
∂s s=0 1 − 2p
(7.21)
whose divergence at p = pc = 1/2 signals the ferromagnetic phase transition. For
p > pc , P (h) loses normalization because a finite fraction of the cavity fields flow to
154
infinity, just as in a percolating cluster distribution. Indeed, the divergent cavity fields
are precisely those attached to spins in percolating clusters. Because these spins are
connected to the positively biased boundary and the temperature is effectively zero,
they spontaneously magnetize to M = 1 = tanh(∞). This provides the spontaneous
magnetization critical exponent:
(1 − p)2
M (H = 0, J = ∞, p) = 1 −
∝ (p − pc )1
2
p
from which we read β 0 = 1 in accord with the discussion following Eq. (7.9).
We now concentrate on the p ≤ pc = 1/2 region at finite H. Consider the
magnetization per spin
M (H) =
*
tanh(H +
q
X
+
(7.22)
u(hi + H, Ji ))
i=1
,hi
which is obtained by averaging over the disorder and the distribution of h. It is
straightforward to show that for small h and H we obtain the results of Equation
(7.15). Indeed:
M (H) =
*
H+
q
X
i (hi + H)
i=1
+
,h
+ O H3
= (1 + 3p)H + 3p hhi + O H 3 ;
(7.23)
(7.24)
Now substitute (7.21) and simplify
M (H) = H
(1 + p)
+ O H3 ,
1 − 2p
in accordance with (7.15).
155
(7.25)
We now reconstruct the full probability distribution P (h) exactly. We expand the
function g(s) as a series in e−sH
g(s) =
X
αn e−snH
(7.26)
n≥0
which defines the coefficients αn . P (h) is now given by the inverse Laplace transform,
P (h) =
X
n≥0
αn δ(h − nH).
(7.27)
The αn are given by the series expansion of the square root in (7.20):
αn =
(4p(1 − p))n+2
(−1)n+1 Γ(3/2)
.
2p2
Γ(n + 3)Γ(−n − 1/2)
(7.28)
That is, P (h) is a comb of delta functions at integer multiples of H with a decaying
envelope. The large n behavior of the envelope is
αn =
4(1 − p)2 −3/2 −A(p)n
√
n
e
+ ... ,
π
(7.29)
where A(p) is defined in (7.13). It is not surprising that the same asymptotics governs
both P (h) and Wn .
Finally, to connect directly with the previous section let us compute the exact
magnetization of a spin as a function of applied field H. Evaluating the cavity magnetization equation (7.22) using the cavity field distribution (7.27), we find
q ∞
X
X
q j
q−j
M (H) =
p (1 − p)
j
m ,··· ,m
j=0
1
j =0
αm1 · · · αmj tanh ((j + 1 + m1 + · · · + mj )H)
(7.30)
which naturally expands as a series in tanh nH. At q = 3, we can evaluate all the
coefficients in this series to find that indeed they are identical to the coefficients
156
nWn−1 of Eq. (7.11). Thus, at J = ∞ the cluster series and the cavity method
produce identical results for the magnetization.
Having established the equivalence of the two solutions, we now return to the
analysis of the series for the magnetization. In the following two sections we will
extract the critical behavior near p = pc and, by analytic continuation to imaginary
H, the GM singularity in the density of LY zeros.
7.6
Integral representation for the magnetization
Despite its simplicity, the expansion of Eq. (7.11) is an exact result for the magnetization which can be analytically continued to imaginary values of the magnetic field.
However, the representation of M (H) as a sum in (7.11) is not best suited for this
purpose. An integral representation would be preferable. To obtain it we write the
Laplace transform of the function tanh x
f (s) =
Z
∞
dxe−sx tanh x
0
1
=
2
s
2
− −ψ
+ψ
s
4
2+s
4
,
(7.31)
where ψ is the digamma function. The function f (s) has simples poles only at the
negative even integers and thus we can invert the transform and write
tanh x =
Z
B
ds sx
e f (s),
2πi
(7.32)
where B is any Bromwich path lying to the right of all poles of f (s), that is to the
right of the negative real axis. Inserting into (7.11), we can invert sum and integral,
provided
4p(1 − p)esH < 1.
157
(7.33)
The resulting expression, valid for |arg H| < π/2,
Z
ds
f (s)
B 2πi
X 2(n + 1)!
×
es(n+1)H (p(1 − p))n
(n + 3)!n!
n≥0
M (H) = 3(1 − p)
3
can be written in closed form by performing the sum. This amounts to calculating
the derivative of the generating function of the probability Wn . For the Bethe lattice
the generating function is4
φ(x) =
X
W n xn
n≥0
= −2(1 − p)3 x−3 (8(1 −
p
p
1 − ξ) + 4(2 1 − ξ − 3)ξ + 3ξ 2 )
(7.34)
where ξ = 4p(1 − p)x has been defined for convenience. By means of this function we
can perform the sum inside the integral to obtain
Z
3(1 − p)3
ds
M (H) =
f (s)e−2sH
3
3
6p (1 − p) B 2iπ
p
× (p(1 − p)esH − 1) 1 − 4p(1 − p)esH + 1 − 3p(1 − p)esH . (7.35)
We simplify this expression by considering the analytic structure of the integrand
(see figure 7.3). The function f (s) has simple poles at the non-positive even integers
(0,-2,-4,...), while the rational expression has a series of square root cuts at s∗n =
A(p)
H
+ i2πn/H. We close the contour with a semicircle at infinity on the right (for
Re s > 0) on the first Riemann sheet. We then deform the contour to coincide with
4
The generating function can be derived for q = 4 as well while for generic q the result is written
in terms of hypergeometric functions for which no explicit rational form seems to exist. We see
below that the fundamental property of this generating function is to have a square-root branch cut
at ξ = 1. This exists for all q > 2 due to a well-known property of the generalized hypergeometric
functions.
158
s
B
Figure 7.3: Analytic structure of integrand of equation (7.35). The function f (s) has
simple poles at non-positive even integers while the rational expression has square
root cuts at s∗n = A(p)+2πin
which we connect to infinity at the right. B labels the
H
undeformed Bromwich path.
the edges of the cuts. At this point only the discontinuity across the cuts contributes
to the final result. For aesthetic reasons we finally shift the value of s by
A(p)
,
H
the
real part of the origins of the cuts.
The resulting expression is
∞ Z ∞
8(1 − p)2 X
M (H) =
dsf (s + s∗n )
πp
n=−∞ 0
1 sH p sH
−2sH
× e
1− e
e − 1.
4
(7.36)
At this point it seems we have traded a sum of functions (7.16) with a series of
integrals that we cannot evaluate. This looks like a step backward in the quest for
a useful result! However, after thinking about the procedure we have performed,
we recognize that this is a Poisson summation-like duality on the original equation
(7.11). The terms in the sum are higher and higher harmonics of the result (this is
particularly evident, as we will see shortly, for imaginary H).
159
The series in n in (7.36) is dual to the series in (7.11) so that when the first
converges rapidly the second does not and vice versa (for H on the real axis). In the
interesting regime, close to the percolation threshold (7.11) converges slowly and the
first term (n = 0) of (7.36) gives the leading term in the expansion in (p − pc ) and
H → 0.
Let us now see how we can recover Eq. (7.16) in a clean way. At the critical point
p = 1/2, we have A = 0 and so s∗n = i2πn/H. For H → 0 all the cuts except that
corresponding to n = 0 go to infinity and we can keep only the n = 0 term in the
series (7.36). Moreover, by expanding the integrand in powers of H we find
Z ∞
√
3√
M (H) '
H
sf (s) + O (H)
π
0
√ Z ∞
3√
π
=
H
x−3/2 tanh x + O (H)
π
2 0
which coincides with Eq. (7.16).
To recapitulate, the magnetization is given by an integral of the discontinuous
part of the generating function φ of the cluster distribution Wn with the Laplace
transform of the function tanh x. For the Bethe lattice the generating function can
be written down explicitly and the calculations can be carried to the end. In the
percolation limit the cut on the real axis gives the greatest contribution to the sum.
7.7
Density of Lee-Yang zeros at J = ∞
In this section we will find the density of Lee-Yang zeros ρ at J = ∞. These are
the zeros of the partition function as a function of the external magnetic field H, for
imaginary H = iθ. Instead of solving the equation Z(H) = 0 directly, we rely on the
relation [24]
ρ(θ) =
1
Re M (iθ + 0+ ).
π
160
To get an idea of how this function looks we recall
+
Re tanh(iθ + 0 ) = π
∞
X
m=−∞
δ(θ −
π
(2m + 1)).
2
Substituting this into (7.15), we find
ρ(θ) =
XX
m n≥0
X
Wn (n + 1)δ((n + 1)θ −
2m + 1
=
Wn δ θ − π
2n + 2
m,n
so the zeros are located at all the
odd
even
π
(2m + 1))
2
(7.37)
rational multiples of π, with multiplicities given
by the Wn ’s5 . This is a singular distribution with an accumulation point at θ = 0:
our task is now to smooth it by using the Poisson-dual integral representation (7.36)
obtained in the previous section.
For imaginary H, the expansion over the cuts becomes an harmonic expansion (see
Fig. 7.4) of ρ. Selecting the term with n = 0 in (7.36), gives the function smoothed
to the lowest degree:
8(1 − p)2
ρsm (θ) =
π2p
Z
0
∞
p
1 sθ −2sθ
sθ
ds e − 1 1 − e
e
Im f (−is − iA(p)/θ + 0+ ).
4
(7.38)
This expression simplifies since from the definition of f
+
Im f (−iz + 0 ) =
Z
∞
dx sin(zx) tanh x =
0
π
.
2 sinh πz/2
(7.39)
5
Usually, disorder averaging generates a smooth density of LY zeros but not so in the infinite
coupling limit of the diluted ferromagnet. This is because the thermodynamics only depend on the
size of clusters and not their shape and each cluster of size n contributes zeros precisely at odd
multiples of π/2n.
161
Figure 7.4: The density of Lee-Yang zeros as a function of the imaginary field θ
at p = 1/4. The upward bending smooth curve (red) is the smoothed ρsm ; the
oscillatory curve (orange) is the sum of the first three harmonics in (7.36) (terms
n = ±1, ±2, ±3) and the downward bending smooth curve (blue) is −ρsm . The
figure suggests (in agreement with the discussion in the text) that the sum of all
the harmonics (with n 6= 0) builds a sum of delta functions Eq. (7.37) minus ρsm in
(7.40).
Figure 7.5: The density of Lee-Yang zeros as a function of the imaginary field θ at
p = 1/10 (left) and at criticality p = 1/2 (right).
So we find the smoothed density of LY zeros
4(1 − p)2
ρsm (θ) =
πp
Z
0
∞
p
1
sθ
ds esθ − 1 1 − e
4
sinh
e−2sθ
.
π
(s
+
A/θ)
2
(7.40)
The different profiles for this density can be seen in Fig. 7.5. Here the GM
phenomenon is evident: even at p < pc = 1/2, ρsm (θ) is strictly positive for any
162
non-zero θ; there is no gap in the distribution. This effect is due to the presence of
rare large clusters. The asymptotic expansion of the density at small θ can be found
by expanding the integrand to leading order in θ:
Z
4(1 − p)2 ∞ √
3
1
ρ(θ) '
ds sθ
π
π
1
s
πp
4 2 e 2 e 2 A/θ
0
√
3 2(1 − p)2 √ − π A/θ
'
θe 2 .
4π 2 p
(7.41)
This expansion is uniformly valid at the point A = 0, which is p = pc , where it shows
the critical square root cusp in the magnetization. However, let us remark that Eq.
(7.40) is the smoothed part (in the sense of distributions) for all values of θ and not
only for small θ.
Let us now make a few qualitative remarks on the asymptotic expansions for
M (H) and ρ which apply in principle to all lattices. From the integral representation
(7.35) (see Fig. 7.3) we observe that there is no Stokes phenomenon for Re H > 0.
That is, the asymptotic approximation for M (H), equation (7.15) to higher order,
ML (H) =
L
X
k=0
ak H 2k+1 (n + 1)2k+2 + O H 2L+1
(7.42)
(where ak ’s are the coefficients in the series expansion for tanh x) is valid for all
|arg H| < π/2.
Naively, substituting H = iθ + 0+ term by term into this expansion, we obtain
a purely imaginary result for any L. Since ML (H) is odd, we might speculate that
ρ = 0. However, the expansion (7.42) is only asymptotic, since (n + 1)k ∼ k!e−kA .
This means that we cannot take L → ∞ but must instead truncate the series at the
L ∼ H/A where the remainder is smallest. The remainder is never actually zero but
it is exponentially small in 1/|H|. A good quantitative approximation can be found
by using the “terminant” [35] of the asymptotic expansion. The terminant is indeed
163
∝ e−A/|H| and it is not purely imaginary for H = iθ + 0+ . Thus, as a general rule we
expect the real part of the terminant of the asymptotic expansion of M (H) represents
the density of LY zeros in the subcritical region.
7.8
Numerical results
All of the statistical observables of the diluted ferromagnet can be derived from the
cavity field distribution P (h). This distribution is the fixed point of the flow of the
cavity equation (7.2). While we have analytically extracted many of the results of this
chapter, we rely on population dynamics for many of the finite coupling results. See
Sec. 4.6 for a general introduction to population dynamics. In practice, population
dynamics for the diluted ferromagnet converges quickly deep in either the ferromagnetic or paramagnetic phase but slows near the phase transition. We illustrate some
typical results below for q = 3.
Even at the percolating critical point, when the expected cavity distribution develops a long tail and divergent moments, the procedure works. Figure 7.6 shows the
numerically determined cavity field distribution for the p = 0.5, J = ∞ critical point
with applied field H = 1. As noted in Sec. 7.5, the exact solution is a comb of delta
functions at h = nH, n ∈ N with weights decaying asymptotically as a power law
αn ∼ n−3/2 . The numerical solution concentrates on integer fields with a power-law
tail consistent with the exponent −1.5.
In general, the form of the cavity field distribution at finite p and J is only
obtainable numerically. Figure 7.7 shows the numerically determined distribution at
small applied field H on three different points in the p−J plane near the p = 0.75, J =
0.80 critical point. These distributions are typical and illustrate the dramatic increase
in susceptibility on the ferromagnetic side of the phase boundary.
164
Figure 7.6: Cavity field distribution at p = 0.5, J = ∞ critical point with H = 1.
This log-log plot shows agreement with the asymptotic form P (h) ∼ h−3/2 found in
the exact solution.
Figure 7.7: Cavity field distributions as function of p near the p = 0.75, J = 0.80
critical point at small applied field H ≈ 10−4 . The solid (blue) curve is in the
paramagnetic regime, the dashed (green) is in the critical phase and the dot dashed
(red) in the ferromagnetic one. Notice the dramatic increase in the response of the
cavity field distribution to the applied field on the ferromagnetic side of the phase
boundary.
Finally, Figure 7.8 confirms numerically the critical exponents derived using the
moment flow analysis of Sec. 7.3 and the exact solution of Sec. 7.4.
165
Figure 7.8: Critical magnetization in an external field along the J = ∞ line. The
slope of the log-log curves indicates the critical exponent δ associated with each of
the four points p = 0.25, 0.5, 0.75, 1 along the phase boundary. We find δ = 1 for
p = 0.25 < pc = 0.5, δ = 2 for p = 0.5 and δ = 3 for p = 0.75 and 1.
7.9
Summary
The diluted Ising ferromagnet on a Bethe lattice is a tractable model that beautifully
illustrates many of the key physical features of short-ranged disordered systems. In
this chapter, we have attempted to present a unified analysis of the model in the
framework of the cavity method, from which we derive both well-known elementary
results about its phases and non-trivial features such as GM singularities and the
infinite coupling critical exponents.
In particular, the ferromagnetic phase boundary lies in the mean-field universality
class (δ = 3) at any dilution above the percolation threshold. At this threshold
however, the ferromagnetic critical coupling diverges (J → ∞) and our closed form
solutions for the cavity distributions in this limit reveal that the critical behavior is
governed by the percolation of the underlying lattice (δ = 2). Linear stability analysis
of the flow of the cavity moments near criticality naturally reveals the Lyapunov
exponents and the associated correlation depth of the stable phases.
Furthermore, at infinite coupling we have explicitly exhibited the essential
Griffiths-McCoy singularities in the magnetization for all p < pc , where there is no
166
spontaneous magnetization. By a harmonic resummation of the exact magnetization,
we found the smoothed density of LY zeros exactly and conjectured its relation to the
real part of an appropriate terminant of the asymptotic series for the magnetization.
167
Chapter 8
There are no Goldstone bosons on
the Bethe lattice
8.1
Introduction
In this chapter we study the quantum unfrustrated (ferromagnetic) problem on the
Bethe lattice analytically.1 For technical reasons, we do this in the form of the spherical model for a scalar field but it is essentially also the large Nf limit of the nearest
neighbor O(Nf ) quantum rotor model on the Bethe lattice. We find some surprising
results from the perspective of quantum phase transitions on Euclidean lattices. First,
we find that the critical point is marked by a single global mode (or Nf modes in the
O(Nf ) interpretation) descending in energy to become degenerate with the ground
state while all other states remain a finite distance away in energy. Second, no local
operators couple to this mode in the thermodynamic limit so that all local correlation
functions exhibit a gap at criticality. Alternatively, once the thermodynamic limit
has been taken, the response to a field on any finite subvolume is bounded even if
that volume is taken to infinity. Third, this state of affairs persists into the broken
1
Chapter based on work with S. A. Parameswaran and S. L. Sondhi [79].
168
symmetry phase in that while there are 2 (or O(Nf ) worth of) degenerate broken symmetry states as required by the global symmetry, there are no Goldstone bosons and
all local correlators remain gapped. We believe our results are robust to moving away
from the spherical/Nf = ∞ limit. Indeed, the Nf = 2 problem is the particle-hole
symmetric transition in the Bose-Hubbard model studied using the cavity method
by Semerjian et al [110] whose results imply that a macroscopic superfluid density
develops at the quantum phase transition (QPT) without closing the excitation gap
in the single site correlation function.
These results should surprise the reader. On the face of it, the lack of Goldstone behavior confounds the standard intuition regarding broken symmetry phases.
Aficionados of classical models on the Bethe lattice may counter, rightly, that their
phase transitions are accompanied by exponential rather than power law correlations,
as this is sufficient for the classical susceptibility to diverge. However, if one computes
the quantum susceptibility – by which we mean the response to an infinitesimal field
applied over a subvolume whose size is taken to infinity after the thermodynamic
limit – it remains finite, because the matrix elements in question do not couple to the
global mode. The analogous susceptibility of the classical model diverges to signal
the phase transition.
We turn now to a brief discussion of what we mean by the Bethe lattice as it
plays an important role in the following. The traditional, and simplest, way to define the Bethe lattice is through a sequence of Cayley trees. The Bethe lattice then
corresponds to the interior of a large tree where each site is connected to exactly
z neighbors. This construction makes clear that the number of sites within a fixed
distance of a given site scales exponentially with distance and thus connects, qualitatively, with the infinite dimensional limit of hypercubic lattices that is also often
invoked in the context of making self-consistent treatments exact. Unfortunately, the
boundary of a finite Cayley tree is always a finite fraction of the bulk whence the
169
choice of boundary conditions can complicate the thermodynamic limit [39]. Indeed,
the difference between a ferromagnet and a glass can be phrased solely in terms of
boundary conditions. To get around this problem, one can proceed differently and
consider instead the ensemble of z−regular graphs with N nodes. This consists of all
graphs where each site is connected to exactly z nearest neighbors. Picking a graph
at random from this ensemble in the N → ∞ limit yields an alternative definition
of the Bethe lattice which is known to be perfectly satisfactory for the classical ferromagnetic problem [65] and that is what we shall use in this chapter. At finite N ,
members of the ensemble contain loops of characteristic size log N and the graphs are
not entirely homogeneous. However, homogeneity is restored in the limit N → ∞ as
the loop size diverges.
With this definition in hand, we can comment on the feature of the Bethe lattice
that brings about the surprising features we report: the boundary of any subvolume
of the lattice is always a finite fraction of the whole whence (arbitrarily) localized
excitations cannot be made low energy. In more formal terms, the Bethe lattice
has what is known to graph theorists as a positive Cheeger constant h – that is
the minimal ratio of boundary to bulk of its nontrivial subgraphs. A theorem – the
Cheeger bound – guarantees that the Laplacian has a gap of at least h2 /2 between
the uniform Perron-Frobenius ground state and the rest of the spectrum. Moreover,
this property is shared by a large collection of so-called expander graphs [63], of much
interest in quantum and classical information theory, and we expect our qualitative
results to be shared by models on any such graph.
In the balance of the chapter, we first introduce and solve the classical spherical
model on the Bethe lattice which introduces most of the ingredients needed in a simpler setting. This classical model on the Bethe lattice has been solved previously [28]
by a different technique. We next introduce and solve the quantum spherical model
which exhibits the features we described above. (We note that the quantum spherical
170
model on Euclidean lattices has been solved by Vojta [118].) We end with a discussion
and comments on the generality of our results.
8.2
The classical spherical model
The classical spherical model on a z-regular graph G with N nodes is defined by the
Hamiltonian
H=
subject to the global constraint that
by
1X
φi Lij φj
2 ij
P
(8.1)
φ2i = N . Here, the graph Laplacian is given
i
Lij = −Aij + zδij
(8.2)
A is the adjacency matrix of G, and the overall shift of z ensures that H is positive
semidefinite. Alternatively, we can view this choice of the Hamiltonian as the one
which gives a system of coupled oscillators with nonnegative frequencies.
8.2.1
Phase transition
The partition function of the model (at inverse temperature β = 1/T ) is given by the
expression
Z(β) =
Z Y
−βH[φ]
dφi e
i
δ
X
φ2i
i
−N
!
(8.3)
Representing the delta function by a Lagrange multiplier λ and performing the Gaussian integral over the φi , we obtain the effective action
Z(β) =
Z
dλe−N [ 2N Tr log{βL−2iλ1}+iλ]
1
171
(8.4)
As N → ∞, this integral may be performed by steepest descent. The effective
propagator for the φ field is given by (defining −µ/T = 2iλ):
hφi φj i = T (L + µ1)−1
ij
and the constraint that
(8.5)
P 2
hφi i = N gives the self-consistency condition
i
N = Tr[T (L + µ1)−1 ]
(8.6)
We may rewrite this in terms of the eigenvalues α of the graph Laplacian
1
1 X
1
=
T
N α α + µ(T )
(8.7)
where we have explicitly written µ(T ) to emphasize that self-consistency forces µ to
depend on temperature. Throughout the chapter, we use indices α, β to label the
modes of the Laplacian L with energy α and eigenvector uαi and i, j to refer to sites
of the lattice.
The spectrum of z-regular random graph Laplacians and the related problem of
hopping on the Bethe lattice have been extensively studied [85, 30]. The spectrum
√
√
consists of a continuum of states in the interval [z − 2 z − 1, z + 2 z − 1], with the
density of states
z
ρc () =
2π
p
4(z − 1) − ( − z)2
z 2 − ( − z)2
(8.8)
However, the above density of states does not include the uniform Perron-Frobenius
√
eigenvector, which has amplitude 1/ N on each site and energy 0. Thus, the full
density of states is
ρ() =
1
δ() + ρc ()
N
172
(8.9)
The existence of this spectral gap, and the uniqueness of the low-lying state is guaranteed by the fact that the z-regular random graphs have a positive Cheeger constant.
We now return to the self-consistency equation Eq. (8.7) and rewrite it in terms
of the density of states Eq. (8.9):
1
1
=
+
T
Nµ
√
z+2
Z z−1
z−2
d
√
ρc ()
+µ
(8.10)
(z−1)
The usual argument for Bose-Einstein condensation now follows: as T → 0, we must
decrease µ in order to satisfy this sum rule. At high temperatures, the thermodynamic
limit can be taken straightforwardly without paying special attention to the uniform
state, and one can satisfy the self-consistency equation. However, since µ cannot
decrease below the lowest eigenvalue of the Laplacian - as this would render the
steepest descent calculation unstable - the smallest that µ can be is zero. This occurs
at a critical temperature Tc given by


Tc = 

√
z+2
Z z−1
√
z−2
(z−1)
d
−1
ρc () 


=
z(z − 2)
z−1
(8.11)
This result for the transition temperature was previously derived in Ref. [28].
Clearly, Tc = 0 for z = 2, and there is no finite-temperature transition on the
chain, as expected. However, Tc is finite for z > 2, and below this temperature,
we cannot satisfy the self-consistent equation in the naive thermodynamic limit discussed above. Rather, we must keep track of the uniform mode in (8.10) and take
−1
T
µ= N
1 − TTc
+ O(N −2 ) for T < Tc .
For comparison, in the O(Nf ) model, the Lagrange multiplier λ would become a
field λi , which at finite N on the random graph need not be homogeneous at the large
Nf saddle point. However, the infinite Bethe lattice is homogeneous and so in the
173
thermodynamic limit we expect to recover the same saddle point as in the simpler
spherical model.
8.2.2
Ordered phase
How does the system behave in the low temperature phase? Consider the correlation
function computed in the eigenbasis of L, namely
T δαβ
α + µ
hφα φβ i =
(8.12)
Below the critical point, the amplitude of the lowest eigenmode is given by
T
hφ0 φ0 i = = N
µ
T
1−
Tc
(8.13)
The thermal occupation of the uniform state is macroscopic, indicative of long range
order in the system. In light of this, we revisit the steepest descent calculation, but
this time treat the lowest mode separately. First, integrate out the N − 1 higher
modes
Z(β) =
=
Z
dλdφ0 e
Z
dφ0 dλ e
iλ(φ20 −N )
Z Y
dφα e
− 12
P
(βα −2iλ)φ2α
α6=0
α6=0
" #
φ2
1 P
−N iλ 1− N0 + 2N
log{βα −2iλ}
α6=0
(8.14)
and now perform the λ integral by steepest descent, giving the self-consistency equation (again µ/T = −2iλ)
φ2
1= 0 +T
N
Z
√
z+2 z−1
√
z−2 z−1
174
d
ρc ()
+µ
(8.15)
√
For T near Tc , µ z − 2 z − 1 so we expand in powers of µ and solve:
1
µ≈
bT
where b =
R z+2√z−1
√
z−2 z−1
φ20
T
+
−1
N
Tc
(8.16)
d ρc()
2 .
Using this solution and approximating the steepest descent value of the integrand
to quadratic order in µ(φ0 ), we find that
Z(β) ≈
Z
dφ0 e−N Vef f (φ0 )
(8.17)
where the effective potential for φ0 is (dropping constants):
1
Vef f (φ0 ) =
2bT 2
"
φ
√0
N
2 4 #
T
1 φ0
√
−1 +
Tc
2
N
(8.18)
Thus, we see that the effective potential has Landau-Ginzburg form, and predicts
√
a mean field symmetry breaking for T < Tc such that hφ0 i ∼ N . That is, hφi i ∼
√
hφ0 i/ N ∼ O(1).
8.2.3
Spatial correlations
We now consider the spatial behavior of the Green’s function Gij = hφi φj i. As this is
ultimately a Gaussian theory, the only interesting correlator is the two-point function.
Since we do not know the exact eigenvectors of the graph Laplacian at finite N , we
cannot obtain Gij from Gαβ directly. Instead, we work directly in the thermodynamic
limit of the Bethe lattice and assume that Gij = G(|i − j|) is translation invariant.
In the disordered phase, Gij satisfies the equation of motion
(Lii0 + µδii0 )Gi0 j = −δij
175
(8.19)
Using the ansatz Gij = Ae−|i−j|/ξ , one finds
Gij =
2(z − 1)
p
(z − 2)(z + µ) + z (z + µ)2 − 4(z − 1)
!|i−j|
p
(z + µ) − (z + µ)2 − 4(z − 1)
×
2(z − 1)
(8.20)
which, as µ → 0+ at the phase transition, reduces to
Gij =
z−1
(z − 1)−|i−j|
z(z − 2)
(8.21)
This agrees with the well-known correlation length ξ = 1/ ln(z − 1) of the Bethe
lattice at criticality and corresponds to a diverging global susceptibility.
Since the uniform mode has vanishing weight in the disordered phase, it makes no
contribution to the Green’s function for T > Tc . In the low temperature phase, the
uniform mode has macroscopic occupation corresponding to a non-zero value of hφi i.
In this case, the expression (8.12), calculated at µ = 0, corresponds to the connected
correlation function Gcij = hφi φj i − hφi ihφj i throughout the low temperature phase.
That the correlation length sticks at its critical value and the susceptibility remains
divergent throughout the condensed phase is the Bethe lattice analogue of the classical
Goldstone theorem, which arises quite naturally when this model is viewed from the
large Nf point of view.
8.3
The quantum spherical model
The quantum spherical model is given by the Hamiltonian [118]
1 X 2 1X
H= g
π +
φi Lij φj
2 i i 2 ij
176
(8.22)
with canonical commutation relations [φi , πj ] = iδij , and the mean spherical constraint:
X
hφ2i i = N
(8.23)
i
As discussed in Ref. [118], the partition function can be rewritten in functional
integral form as
Z(g, β) =
+
Z
dλ
Y
i
( Z
dφi (τ ) exp −
β
dτ
0
"
1 X
φ̇i (τ )2
2g i
!#)
X
1X
φi (τ )Lij φj (τ ) + iλ
φ2i (τ ) − N
2 ij
i
(8.24)
where τ is an imaginary time parameter. Transforming to frequency space and expanding in eigenvectors of L,
Z(g, β) =
Z
dλ
Y
dφα (ω)e
− 21
P
ω,α
h 2
i
φα (ω) β ωg +α −2iλ φα (−ω)−iλN
(8.25)
α
where we sum over discrete frequencies ωn =
2πn
.
β
The self-consistency equation is now given by
1=
T X
1
2
N ω,α ω + + µ
α
g
(8.26)
Performing the frequency summation, we obtain
√
g
1 X
1 p
√
1=
coth
β g(α + µ)
N α 2 α + µ
2
(8.27)
This result was computed directly within the Hamiltonian formalism in Ref. [118].
Note that as g → 0, we recover the classical result Eq. (8.7).
177
8.3.1
Quantum phase transition
At T = 0, the self-consistency equation reduces to
1
1 X
1
√
√ =
g
N α 2 α + µ
=
1
√ +
2N µ
√
z+2
Z z−1
√
z−2 z−1
ρc ()
d √
2 +µ
(8.28)
Following a similar argument as in the classical case, we find that the system has a
quantum critical point at the critical coupling gc , defined by
gc−1/2 =
√
z+2
Z z−1
√
z−2
ρc ()
d √
2 (8.29)
(z−1)
This integral is convergent, and can be done numerically to find the precise value of
gc . Again, for g < gc we must be careful regarding the thermodynamic limit and take
√
√ −1
p
g
µ=
1 − g/gc
2N
(8.30)
in order to determine the macroscopic occupation of the zero mode. As expected,
the quantum spherical model undergoes a Bose condensation transition at the critical
coupling gc .
8.3.2
Correlations
The imaginary time Green’s function at zero temperature is most easily computed by
taking a limit from finite temperature. From the partition function Eq. (8.25), we
178
find
Gαβ (τ )
=
X
ω
=
Z
T δαβ
eiωτ
+ α + µ
dω
eiωτ
2
2π ωg + α + µ
√
√
g
δαβ √
e− g(α +µ)|τ |
2 α + µ
−→ δαβ
T →0
ω2
g
(8.31)
Each mode α decays exponentially in imaginary time at a rate corresponding to the
gap to exciting that mode. Thus, the spectral response of the time-ordered propagator
p
in real frequency space is precisely a delta function at ω = g( + µ).
Now we are in a position to understand the peculiarity of the quantum conden-
sation transition on the Bethe lattice. The single site Green’s function is given by
transforming Gαβ back to position space using the eigenmodes uαi of the graph Laplacian:
√
√
X
p
g
g
1
√
α α
Gii (ω) =
ui ui √
δ(ω − g(α + µ))
√ δ(ω − gµ) +
N2 µ
2 α + µ
α>0
Without detailed knowledge of the eigenvectors of the Laplacian but merely its density
of states, this formula already allows us to sketch the spectral response of local single
particle excitations as in Fig. 8.1. As expected, the quantum phase transition is
signaled by the continuous closing of a spectral gap to a uniform mode. The unusual
feature is that this state is isolated from the remainder of the spectrum and its weight
vanishes as 1/N in the disordered phase. Meanwhile, in the ordered phase, the weight
√
p
g
1
√ = 1 − g/gc
N2 µ
is finite, reflecting the condensation into the uniform state.
179
(8.32)
Figure 8.1: Support of Gij (ω) at zero temperature for z = 3. The isolated (blue)
line indicates the contribution of the uniform mode which vanishes as 1/N above gc
(dashed) but is finite below gc signaling the long-range order (solid). Obtained from
numerical inversion of Eq. (8.28).
The general spectral features of correlations sketched above may be supplemented
in detail by assuming homogeneity of the infinite Bethe lattice and exploiting our
knowledge of the classical spherical model. The frequency resolution of the zero
temperature Green’s function Gij (ω) is given by the mode sum
Gij (ω) =
X
α
uαi uαj
ω2
g
+ α + µ
(8.33)
which has the same form as the mode sum for a classical model at chemical potential
µ+
ω2
.
g
Thus, we use the classical Green’s function in Eq. (8.20) and make this
substitution for µ. The formal expression is rather unenlightening but there is much
information in the pole structure (see Fig. 8.2). The most important feature is a
pair of vertical square root branch cuts on the imaginary axis corresponding to the
bulk spectrum. These never pinch off the real axis as g is varied; rather, the closest
branch point sets the dominant decay rate, which agrees exactly with the previous
discussion.
At finite N the branch cuts break into lines of poles and in addition one should
recall the contribution of the uniform mode. This provides a pair of isolated simple
180
Figure 8.2: Analytic structure of Gij (ω) in the complex plane. Wiggly lines (green)
indicate branch cuts. The two crosses (red) indicate the poles due to the zero mode
with residue 1/N . The following formulae for cut locations are given at µ = 0
(criticality): The hyperbolic
are given by Re [ω]2 − Im [ω]2 = −zg. The vertical
q cuts √
cuts extend between ±i g(z ± 2 z − 1). These cuts control the slowest decay since
we can deform the contour past the uniform mode pole in the thermodynamic limit.
√
poles at ±i gµ with residue 1/N in the disordered phase. As g → gc+ , these poles
pinch off the real axis corresponding to the closing of the gap and the phase transition.
In the condensed phase, these poles have non-vanishing residue indicating the long
range order in imaginary time.
Finally, we note that the static susceptibility χ of the quantum model to a global
field (applied before the thermodynamic limit) is simply given by the zero frequency
lattice sum of Gij (ω), which is equivalent to summing Eq. (19) over the lattice. This
susceptibility diverges throughout the broken symmetry phase, as usual. However,
taking a large subvolume and computing its susceptibility after the thermodynamic
limit gives a convergent result, reflecting the absence of Goldstone bosons in the
spectrum.
8.4
Discussion
There are two salient qualitative features of the spherical model condensation transition on the Bethe lattice that we have derived above: (1) despite its vanishing weight
181
in the response to local excitations, there is a global mode closing the many-body
gap in the disordered phase. Alternatively, this can be viewed as a ‘midgap’ mode
driving the transition, where by midgap we mean in the middle of the gap to local
excitations. (2) It is via macroscopic occupation of this mode that long range order
develops in the symmetry-broken phase. However, local excitations remain gapped.
In fact, not only are there no Goldstone bosons, the spectral response of any operator
with bounded support will remain gapped.
We believe these features are quite general to unfrustrated transitions on the Bethe
lattice and expander graphs. For one thing, as already noted in the introduction, our
results are entirely consistent with those of Ref. [110] on the Bose-Hubbard model
wherein the particle-hole symmetric transition is the O(Nf ) model at Nf = 2. This
indicates that the features we have found at Nf = ∞ are robust throughout the
range of O(Nf ) models and not merely artifacts of the spherical model limit. We
believe this claim will be susceptible to a proof in the 1/Nf expansion. Furthermore,
the results should apply to sufficiently weakly disordered ferromagnetic models, for
which the disorder does not close the mode gap.
More intuitively, these results should be stable to the inclusion of self-interactions
of the scalar field beyond the self-consistent Gaussian theory treated here. First,
at the critical point, weak interactions cannot close the gap between the soft global
modes and other massive modes; indeed, they typically cause the low energy density of
states to decrease even further. Second, the usual heuristic argument for a Goldstone
mode is seen to lead to a gap as follows: consider the simplest case of an O(2) = U (1)
broken symmetry in the broken phase. In the usual fashion we derive an effective
Lagrangian for the phase fluctuations in the symmetry-broken phase by freezing out
modulus fluctuations in |hφi|:
Leff ≈
X 2 1
X
1
|hφi|2
θ˙i + |hφi|2
θi Lij θj
2g
2
i
ij
182
(8.34)
This is a quadratic action for a massless free field θ, which would lead to gapless behavior of local excitations were the Laplacian gapless. However, the Laplacian spectrum
on expander graphs is gapped to all modulated excitations. While the global rotation
is still, necessarily, of zero energy, it no longer follows that there exist local excitations
of arbitrarily low energy. This corresponds to a breakdown of our intuitive understanding of generalized rigidity. Quite generally then, we conjecture that there is an
‘anti-Goldstone’ theorem that applies to symmetry breaking transitions on expander
graphs.
We should also note that the fully connected models for which mean field theory is
exact exhibit similarly gapped behavior, even after the usual rescaling for extensivity.
Consider the fully connected Heisenberg ferromagnet whose Hamiltonian is H =
P
2
. Trivially, the gap to the first excited state is J and
−J/N ij Si · Sj = −J/N Stot
does not vanish in the thermodynamic limit. The fully connected graph does not have
bounded coordination so is not technically an expander graph, however the surface
to volume ratio is, nonetheless, very large.
Clearly there are many subtle issues that may invalidate the usual physics lore
when models are considered on non-Euclidean graphs. As quantum models are studied more extensively in the context of quantum complexity theory and on random
networks, an enhanced understanding of universal symmetry breaking properties will
potentially play an important role in their analysis.
183
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