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M AT R I X P R O D U C T S TAT E S F O R L AT T I C E G A U G E T H E O R I E S
kai zapp
October 2015
Kai Zapp: Matrix Product States for Lattice Gauge Theories, © October
2015
supervisors:
Jun.-Prof. Román Orús
Univ.-Prof. Harvey B. Meyer
ABSTRACT
In this thesis the matrix product states formalism is used to calculate
the chiral condensate in the massive 1-flavour Schwinger model for
different fermion masses. To this end, we use the one-site infinite
density matrix renormalization group algorithm applied on gauge
invariant matrix product states. The results obtained are in agreement
with previous studies and can be seen as a proof of concept that an
matrix product ansatz can describe the relevant physical states in a
Hamilton lattice gauge theory in (1+1)D.
iii
CONTENTS
i introduction
1
1 introduction and motivation
3
2 basic concepts
5
2.1 Entanglement
5
2.1.1 Entanglement and the EPR Paradox
5
2.1.2 Entanglement and Quantum Information Theory
6
2.1.3 Bipartite Entanglement and the Schmidt Decomposition
7
2.1.4 The Reduced Density Matrix
7
2.1.5 Entanglement Entropy
8
2.1.6 Entanglement in Quantum Many-Body Systems
2.2 The Variational Principle
10
ii tensor network theory
13
3 tensor networks
15
3.1 Why Tensor Networks
15
3.2 Tensors, Tensor Networks and Graphical Notation
15
3.3 Tensor Network Representation of Quantum Many-Body
States
17
3.4 Matrix Product States (MPS)
19
3.5 Construction of an MPS
21
3.6 Matrix Product Operators
26
3.7 Ground State Calculations in one Dimension
27
3.7.1 Infinite Time Evolving Block Decimation (iTEBD)
3.7.2 Infinite Density Matrix Renormalization Group
(iDMRG)
29
4 the ising model in a transverse field
39
4.1 Ground State Properties
39
4.2 Results of the iTEBD Calculations
41
iii the schwinger model
45
5 the schwinger model
47
5.1 The Schwinger Model as a Lattice Field Theory
5.2 Chiral Symmetry and Chiral Condensate
50
6 calculation of the chiral condensate
53
6.1 Gauge Term vs. Thermodynamic Limit
53
6.2 iDMRG with Gauge Invariant MPS
55
7 conclusion and outlook
67
a appendix
69
a.1 The Singular Value Decomposition
69
9
27
48
v
vi
contents
a.2 Continuum Extrapolations of the Subtracted Chiral Condensate
70
bibliography
75
LIST OF FIGURES
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Figure 23
Figure 24
Figure 25
Figure 26
A neutral pion at rest decays into an electronpostitron pair.
5
The entanglement entropy between A and B
scales with the size of the boundary ∂A between the two subsystems.
10
The quantum many-body states obeying an area
law for the scaling of entanglement entropy
correspond to a tiny manifold in huge manybody Hilbert space.
10
Graphical notation of tensors
16
Tensor networks
16
TN diagram: Trace of six matrices
17
TN representation of a quantum many-body
state
18
Matrix Product State: OBC
19
Matrix Product States: PBC and thermodynamic
limit
20
Canonical form of an MPS
21
Canonical Form: Expectation value of a singlesite observable
22
Left- and right-canonical form
22
Mixed canonical form: Single-site expectation
value
22
Successive Schmidt Decomposition
24
Onion-like structure of the Hilbert space.
24
MPS and the Valence Bond Picture
26
Matrix Product Operator
26
MPO acting on MPS
27
iTEBD: Application of UAB on an infinite MPS
with 2-site translational invariance
29
iTEBD: Detailed update process
30
Transformation of a tensor into a vector and
effective Hamiltonian
31
iDMRG: Approximative environments and effective Hamiltonian
34
iDMRG: odd step
34
iDMRG: even step
35
2-site iDMRG
36
Groundstate energy for the transverse Ising model:
Exact and iTEBD calculation
42
vii
Figure 27
Figure 28
Figure 29
Figure 30
Figure 31
Figure 32
Figure 33
Figure 34
Figure 35
Figure 36
Figure 37
Figure 38
Figure 39
Figure 40
Figure 41
Figure 42
Figure 43
Figure 44
Transverse Ising Model: Error in ground state
energy
42
Transverse Ising Model: Magnetization mz for
different bond dimensions
43
Imaginary Time Evolution with an MPO
54
MPO of the local Schwinger model Hamiltonian
57
Schwinger Model with one-site iDMRG
59
Computed chiral condensate for m/g=0.25
60
Subtracted chiral condensate for m/g=0.25
61
Subtracted chiral condensate for m/g=0.25. Focus on small lattice constants.
61
Chiral condensate: extrapolation in the bond
dimension for x = 100
62
Chiral condensate: extrapolation in the bond
dimension for x = 500
63
Continuum extrapolation of the chiral condensate for m/g = 0.25
63
Continuum extrapolation of the chiral condensate for m/g = 0.25
64
Continuum extrapolation of the chiral condensate for m/g = 0.
70
Continuum extrapolation of the chiral condensate for m/g = 0
71
Continuum extrapolation of the chiral condensate for m/g = 0.125
71
Continuum extrapolation of the chiral condensate for m/g = 0.125
72
Continuum extrapolation of the chiral condensate for m/g = 0.5
72
Continuum extrapolation of the chiral condensate for m/g = 0.5
73
L I S T O F TA B L E S
Table 1
Table 2
viii
Simulation parameters in the one-site iDMRG
algorithm.
60
Comparison: subtracted chiral condensate in
the continuum.
64
List of Tables
Table 3
Results: subtracted chiral condensate in the continuum.
68
ix
Part I
INTRODUCTION
1
I N T R O D U C T I O N A N D M O T I VAT I O N
Gauge theories have revolutionized our understanding of fundamental interactions. In particular, the Standard Model of particle physics
which is based on gauge theories is presently the best description
of three of the four fundamental forces: the electromagnetic force,
the strong force and the weak force. Within the framework of the
Standard Model, forces between elementary particles are mediated
by gauge fields corresponding to a particular gauge symmetry.
In a perturbative treatment, one uses series expansions of transition amplitudes to obtain physical predictions. At a pictorially level
the coefficients of these power series expansions can be represented
by the well known Feynman diagrams which can be classified by the
order of the coupling constant of the considered theory. Applied to
the fundamental theory of electromagnetism, namely quantum electrodynamics (QED) this approach was extremly successful, and led to
very precise predictions like the Lamb shift or the magnetic moment
of the electron [38].
However, perturbation theory as a calculational tool fails once interactions become strong, as it is the case for instance in low-energy
quantum chromodynamics (QCD)1 . In this non-perturbative regime,
Lattice QCD, which is based on Monte Carlo evaluation of the discretized Euclidean path integral, has become a most powerful quantitative tool. For example Lattice QCD calculations for the light hadron
mass spectrum have reached an impressive agreement with experimental data [14, 20]. But despite being a highly mature field, as a
Monte Carlo method it suffers from the so-called sign problem which
makes calculations for systems with large fermionic densities computationally inaccessible. Furthermore, the use of Euclidean time instead of real time presents a serious barrier for the understanding of
out-of-equilibrium dynamics. It is therefore of primary importance to
search for tools which overcome these problems, and can serve as a
complementary ansatz for the numerical simulation of lattice gauge
theories.
This thesis deals with so-called tensor network methods as such
an alternative approach. Tensor network states provide an efficient
description of quantum many-body states based on entanglement
properties. Their main limitation is very different compared to other
numerical techniques: it is defined by the amount and structure of
entanglement in quantum many-body states.
1 QCD is the theory of the strong interactions.
3
4
introduction and motivation
We focus here on the application of so-called matrix product states,
which are tensor network states for one dimensional systems, to the 1flavour massive Schwinger model in its formulation as a Hamiltonian
lattice gauge theory. The Schwinger model, or QED in two space-time
dimensions, is a popular toy model for QCD, since both theories share
different features like confinement or chiral symmetry breaking. We
aim to study the model directly in the thermodynamic limit, since
the possibility of applying algorithms for infinite-size systems would
allow us to estimate the properties without the burden of finite-size
scaling effects.
2
BASIC CONCEPTS
2.1
entanglement
The aim of this section is to familiarize the reader with the notion of
entanglement1 .
2.1.1
Entanglement and the EPR Paradox
Quantum mechanics has features that are radically different from
those known from the classical description of Nature. For example,
one may think of superposition of quantum states, interference, or
tunneling. All these well known examples have one thing in common:
they can already be observed in single-particle systems. But there are
further purely quantum-mechanical phenomena that manifest themselves in systems that are comprised of at least two subsystems. Perhaps one of the most interesting and puzzling features of quantum
mechanics is associated to such composite systems, namely entanglement. The characteristics of an entangled state is the fact that the wavefuctions of the individual particles are not well defined. This is fundamentally different to our classical description of Nature, where full
knowledge of the parts is equivalent to full knowledge of the whole
system. In dealing with entangled states we may even reach paradoxical conclusions by using classical lines of thought. A famous example
of such a paradox was formulated by Einstein, Podolsky and Rosen
in 1935 [15]. It rests on the classical assumption of locality, which in
particular states that no influence can propagate faster than the speed
of light. A simplified version of this so called EPR paradox, illustrated
with spin-half particles, was introduced by Bohm and Aharonov [8].
The argument reads as follows: Let us consider the decay of a system
of zero total angular momentum into two spin-1/2 particles. For concreteness, we can think of the decay of a neutral pi meson in its rest
frame into an electron and a positron (Fig. 1).
e
-
!0
e+
Figure 1: A neutral pion at rest decays into an electron-postitron pair.
1 “Entanglement” is the English translation of the German word “Verschränkung”,
which was first introduced by Schrödinger [37].
5
6
basic concepts
Using Clebsch-Gordan coefficients to ensure conservation of angular momentum, we see that the spin vector of the electron-positron
system has to be in the singlet configuration2
1
|ψe− e+ i = √ (|↑− i |↓+ i − |↓− i |↑+ i) .
2
(2.1)
This is a paradigmatic example of an entangled state; the z-component
of the spin of each particle in this state is not well defined. However,
the situation changes when measurements of the spin of one of the
particles are made. For example, by measuring the spin of the electron it becomes either up or down. Furthermore, then also the spin of
the positron is immediately determined. It must be either down or up
respectively. No matter how far electron and positron are apart, the
measurement on one particle has an instantaneous (i.e. faster-thanlight) effect on the other. It was this, in Einstein’s words, “spooky
action at a distance” that led Einstein, Podolsky, and Rosen to the
conclusion that the quantum mechanical description of physical reality in terms of wavefunctions had to be incomplete. In order to save
locality, a number of so called hidden variable theories, which sought
to supplement quantum mechanics, were introduced. Later, however,
in 1964 Bell published his famous theorem (Bell’s theorem) [5], and
proved that these theories must yield to predictions that are inconsistent with quantum mechanics. In particular, he was able to show
that all local hidden variable theories must satisfy Bell’s inequalities.
Since 1982 a great number of experiments confirmed the violation of
Bell’s inequalities, e.g., Refs. [3, 50], demonstrating that Nature itself
is fundamentally nonlocal.
2.1.2
Entanglement and Quantum Information Theory
Over the recent years our understanding and knowledge of entanglement, although far from being complete, made significant progress.
For example, in the context of quantum information theory entanglement was identified as a useful resource, like energy, which can be
used to perform tasks that could not be achieved with classical states.
Among the applications are quantum cryptography [17], superdense
coding [6], and quantum teleportation [7].
Furthermore, it has become clear that concepts and methods of
entanglement theory which originally emerged from quantum information theory, can lead to new insights in the context of many-body systems. Important examples are the so-called area-laws for the entanglement entropy, i.e. laws that characterize entanglement in physically
relevant many-body states. Since these area-laws support the theoretical framework of the tensor network methods used in this thesis, we
2 Sometimes this state is also called a Bell state, or EPR pair.
2.1 entanglement
will discuss them in subsection 2.1.6. But first, we have to introduce
some basic notions.
2.1.3
Bipartite Entanglement and the Schmidt Decomposition
Let us start with a formal definition of entanglement. Here we focus
on bipartite quantum systems AB, i.e. systems which can be decomposed into two different subsystems, A and B. Let |ψAB i be a pure
state living in the tensor product Hilbert space HAB = HA ⊗ HB of
the composite system. It is called separable or product state if it can be
written as the product of pure states, i.e.
|ψAB i = |ψA i ⊗ |ψB i , with |ψA i ∈ HA , |ψB i ∈ HB .
(2.2)
A state that cannot be written as such a product is called an entangled
state. An example of an entangled state was already given in Eq. 2.1.
In general, it is not evident whether a state is separable or entangled. However, for pure states this separabilty problem is easy to handle due to the so-called Schmidt decomposition3 : Let |ψAB i ∈ HAB be
a pure state of a composite system AB. Then there exist an orthonormal basis {|αA i} of subsystem A, and an orthonormal basis {|αB i} of
subsystem B such that
|ψAB i =
χ
X
α=1
λα |αA i |αB i ,
(2.3)
P
2
where λα > 0, χ
α=1 λα = 1, and χ = min {dim HA , dim HB }. The
non-negative numbers λα are the Schmidt coefficients, and χ is called
Schmidt rank. Since the Schmidt basis {|αA i ⊗ |αB i} consists of separable states, the information of entanglement of a given state is encoded
in its (unique) Schmidt coefficients. Pure product states correspond to
those states whose Schmidt decompositions have exact one Schmidt
coefficient. Otherwise, if there are at least two Schmidt coefficients different from zero, the state is entangled. In fact, the number of Schmidt
coefficients is what is called a (discontinuous) measure of entanglement.
The larger χ is, the larger the amount of entanglement in the considered state is.
2.1.4
The Reduced Density Matrix
The Schmidt coefficients can be related to the eigenvalues of the reduced density matrix of either A or B. The reduced density matrix
turned out to be a very usefel tool for the description of the individual
subsystems of a composite system. It describes all the properties or
outcomes of measurements of the considered subsystem, given that
3 The Schmidt decomposition is essentially a restatement of the singular value decomposition (see App. A.1).
7
8
basic concepts
its complement is left unobserved4 . For subsystem A it is obtained by
tracing out the degrees of freedom of B in the joint state |ψAB i, i.e.
ρA = trB (|ψAB i hψAB |) ≡
dim
HB
X
i=1
hiB | (|ψAB i hψAB |) |iB i ,
(2.4)
where trB denotes the partial trace over system B, and {|iB i} is a basis
in B. Note, the partial trace maps operators defined on the Hilbert
space of the composite system to operators defined on the Hilbert
space of the considered subsystem.5 The reduced density matrix for
subsystem B is analogously defined. By choosing the Schmidt basis
to evaluate the reduced density matrices for A and B they can be
diagonalized:
ρA =
ρB =
χ
X
α=1
χ
X
α=1
λ2α |αA i hαA | ,
(2.5)
λ2α |αB i hαB | .
(2.6)
The eigenvalues of ρA and ρB are identical, and are given by the
squares of the Schmidt coefficients. Therefore, the reduced density
matrix is also a useful tool for finding the Schmidt decompostion.
2.1.5
Entanglement Entropy
We already mentioned that the Schmidt rank χ quantifies entanglement in bipartite pure states, and is therefore one possible measure
of entanglement. Here we want to introduce the so-called entanglement entropy of a subsystem, which is the von Neumann entropy of
its reduced density matrix. For subsystem A it is given by
S (ρA ) = − tr ρA log2 ρA .
(2.7)
Or, in terms of the Schmidt basis:
S (ρA ) = −
χ
X
λ2α log2 λ2α = S (ρB ) ,
(2.8)
α=1
where λα are the Schmidt coefficients. Thus the entanglement entropy
is for both subsystems the same. This reflects the intuition that entanglement, as a correlation between A and B, is a common property.
The entanglement entropy is a continuous measure of entanglement.
To illustrate this, let us look again at the singlet state
1
|ψAB i = √ (|↑A i |↓B i − |↓A i |↑B i) .
2
(2.9)
4 For a detailed introduction, including the density matrix formalism, we may refer
the reader to [27].
5 Different to the trace, which maps operators to scalars.
2.1 entanglement
It can be easily seen that this state is already √
in its Schmidt decompostion with Schmidt coefficients λ1 = λ2 = 1/ 2, and Schmidt rank
χ = 2. The entanglement entropy is
1
1 1
1
S = − log2 − log2 = log2 2.
2
2 2
2
(2.10)
It can be shown that the Schmidt rank χ provides an upper bound for
the entanglement entropy, namely
S 6 log2 χ,
(2.11)
and therefore the singlet state is a so-called maximally entangled state.
In contrast to this, for a product state, like e.g.
|ψAB i = |↑A i |↑B i ,
(2.12)
there is always only one Schmidt coefficient with λ1 = 1, and the
entanglement entropy will be S = 0. It is quantum correlations that
make the entropy of reduced states become non-vanishing.
2.1.6
Entanglement in Quantum Many-Body Systems
As already indicated, the study of the entanglement properties has
led to useful insights in the context of quantum many-body systems.
In particular, we want to discuss here qualitatively the so-called arealaw scaling of the entanglement entropy.6
Let us consider a connected subsystem A of a quantum many-body
system and its complement B. Then a natural question might be how
the entanglement entropy between A and B scales. Since the entropy
is an extensive quantity, one could expect that it scales with the volume of the subystem. And indeed, for a quantum state picked at random from the many-body Hilbert state, this will be most likely true.
However, many important Hamiltonians in Nature tend to be local,
with interactions limited to close neighbors. It turns out that this locality of interactions has important consequences. For example, it can
be proven that the low-energy states of gapped many-body Hamiltonians with such local interactions obey an area law. That means the
entanglement entropy of a region of space A and its complement B is
proportional to the area of the boundary separating both regions, see
Fig. 2. That is, low-energy states of gapped models are (much) less
entangled than they actually could be. If we aim to study these states,
this huge constraint on the entanglement properties identifies the relevant, although exponentially small, corner of quantum states in the
many-body Hilbert space, see Fig. 3. This will be key to the understanding of Tensor Network methods introduced in the next chapter.
6 For a detailed review on area-laws the reader may be referred to Ref. [16].
9
10
basic concepts
∂A
B
A
S~∂A
Figure 2: The entanglement entropy between A and B scales with the size of
the boundary ∂A between the two subsystems.
Many-body Hilbert Space
Area-law states
Figure 3: The quantum many-body states obeying an area law for the scaling
of entanglement entropy correspond to a tiny manifold in huge
many-body Hilbert space.
2.2
the variational principle
In this section we review the variational principle. It is the basis for
so-called variational methods such as, for example, the infinite Density
Matrix Renormalization Group (iDMRG) algorithm, which is used in
this thesis.
The variational principle states that the ground state energy E0 of a
system, described by a Hamilitionian H, is always less than or equal
to the expectation value of H in any normalized state |ψi, i.e.
E0 6 hψ| H |ψi , where hψ |ψi = 1.
(2.13)
In other words, the expectation value of H with respect to the chosen
trial wavefunction is always an upper bound for the ground state energy.
To prove this result, we use that the (unknown) eigenstates of H form
a complete set. Therefore, the trial wavefunction |ψi can be written as
X
|ψi =
ck |φk i , with H |φk i = Ek |φk i .
(2.14)
k
The eigenstates themselves are assumed to be orthonormalized, hφk |φl i =
δkl . Hence, we get
XX
X
hψ| H |ψi =
c∗k El cl hφk |φl i =
Ek |ck |2 .
(2.15)
k
l
k
2.2 the variational principle
Since the ground state energy corresponds, by defintion, to the smallest eigenvalue, i.e. E0 6 Ek for all k, it follows
X
hψ| H |ψi > E0
|ck |2 = E0 ,
(2.16)
k
which was to be proven.
Of course, the variational principle per se does not tell us what
kind of trial wave function should be used for a given Hamilitionian H. Successful variational methods rely on educated guesses on
the wavefunction derived from physical insights or intuition.
11
Part II
T E N S O R N E T W O R K T H E O RY
3
TENSOR NETWORKS
This chapter is mainly based on Ref. [28].
3.1
why tensor networks
The description of quantum many-body systems is, in general, an extremly difficult task. This is related to the fact that the size of the
Hilbert space grows exponentially with the size of the given system.
For example, an arbitrary quantum many-body state of a system with
N two-level subsystems already requires the specification of 2N complex numbers. For a classical computer, this inefficient representation
implies both storage and computational problems. On the other hand,
it is well known that a separable state of N qubits1 can be described
with about O (N) paramters. The huge difference, which makes a general state difficult to describe compared to a separable state, lies in the
complexity of quantum correlations, or entanglement. Therefore, one
might intuitively suspect that the low-energy states of local Hamiltoninas which obey an area law (see subsection 2.1.6) can also be described with relatively few parameters compared to a random state
in the many-body Hilbert space. At this point Tensor Networks come
into play. They serve as an efficient parametrization for this small,
albeit fundamental, corner of the Hilbert space.
3.2
tensors, tensor networks and graphical notation
For our purposes, a tensor is defined as a multidimensional array of
complex numbers. The number of indices needed to label a given
tensor is called its rank. According to this definition a scalar (x) is
a
rank-0 tensor, a vector (vα ) is a rank-1 tensor, and a matrix Aαβ is
a rank-2 tensor.
The sum over all possible values of common indices of a set of tensors is called an index contraction. A familiar example is the product
of two matrices
Cαβ =
D
X
Aαγ Bγβ ,
(3.1)
γ=1
1 A qubit is a quantum mechanical two-level system.
15
16
tensor networks
(a)
(b)
(c)
(d)
Figure 4: Graphical notation of tensors: (a) scalar, (b) vector, (c) matrix and
(d) rank-3 tensor
where in this case the contraction is done with respect to the index γ,
with γ = 1, . . . , D. Index contractions can become arbitrary complex,
e.g.
Eαβγδ =
Dρ Dσ Dω
Dν X
X
X X
Aανρω Bβσν Cσρ Dωγδ ,
(3.2)
ν=1 ρ=1 σ=1 ω=1
where the index ι ∈ {ν, ρ, σ, ω} can take Dι different values. Indices
that are not contracted are referred to as open indices.
By a tensor network (TN) we understand a set of tensors whose
indices are partly or wholly connected according to some network
pattern. Eqs. 3.1, 3.2 provide examples of TNs.
At this stage, it is handy to proceed to a graphical notation for tensors and tensor networks by introducing tensor network diagrams. In
these diagrams tensors and their indices are represented by shapes
with outgoing legs, where the number of legs corresponds to the rank
of the tensor, see Fig. 4. A contraction is represented by a connecting
line between two tensors. Examples of contractions in diagrammatic
notation are shown in Fig. 5. Tensor network diagrams serve as a
powerful tool in dealing with TN calculations, since complex equations can be represented in a visual way that is easier to handle, and
that sometimes reveals properties, which are more difficult to see in
plain equations. One such example is the cyclic property of the trace
of a matrix product, which becomes immediately apparent in terms
of TN diagrams, see Fig. 6.
(a)
(c)
β
B
β
(b)
α
E
δ
ν
γ
=
α
σ
ρ
A
ω
D
C
γ
δ
Figure 5: Tensor network diagram notation: (a) matrix product, (b) scalar
product of two vectors, (c) Eq. 3.2 as a TN diagram
3.3 tensor network representation of quantum many-body states
Figure 6: The cyclic property, in this case of six matrices, becomes obvious
in terms of TN diagrams.
3.3
tensor network representation of quantum manybody states
We now turn to explain what we are mainly interested in, namely the
tensor network representation of quantum many-body states. Let us
consider a quantum many-body system of N particles each one with
d degrees of freedom. An arbitrary wave function |ψi that describes
its physical properties can be written as
d
X
|ψi =
i1 ,i2 ,...iN =1
Ci1 i2 ...iN |i1 i ⊗ |i2 i ⊗ · · · ⊗ |iN i ,
(3.3)
where {|ir i} is an individual basis for the single particle states of each
particle r = 1, . . . , N. The dN complex numbers Ci1 i2 ...iN encoding
the many-body wave function can be considered as the coefficients
of a high-rank tensor C with N indices i1 i2 . . . in , where each of the
indices can take d different values. That reflects why quantum manybody systems provide a computational challenge, since already the
description of the wave function by a rank N tensor with O dN
coefficients scales exponentially in the system size, and therefore it is
computationally inefficient. The fundamental idea of tensor networks
is to take the above high-rank tensor C and decompose it into tensors
of smaller rank that are being contracted. Some examples in diagrammatic notation are given in Fig. 7. The number of parameters required
to specify these tensors is much smaller. In fact, the representation of
|ψi in terms of a TN is computationally efficient, since typically it depends on a polynomial number of parameters. In general, the number
of parameters mtot to determine a tensor network is
mtot =
NT
X
m (Ti ) ,
(3.4)
i=1
where m (Ti ) is the number of parameters for tensor Ti , and NT denotes the number of tensors in the TN under consideration. For every
practical tensor network its number of tensors NT is sub-exponential
17
18
tensor networks
(a)
(b)
C
(c)
Figure 7: Tensor network decomposition of coefficient C in (a) an MPS with
open boundary conditions, (b) a PEPS with open boundary condition, and (c) an arbitrary tensor network.
in the system size N, e.g. NT = O (poly(N)). Each of the individual
tensors Ti has a number of parameters given by


rank(Ti )
Y
m (Ti ) = O 
D (ι) ,
(3.5)
ι=1
where the product is taken over all the different indices ι = 1, 2, . . . , rank (Ti )
of the tensor, D (ι) is the number of different values the index ι can
take, and rank (Ti ) is the rank, or equivalently, the number of indices
of the tensor Ti . If we denote with DTi the maximum of all D (ι), then
we have
m (Ti ) = O (DTi )rank(Ti ) .
(3.6)
All in all, the total number of parameters is
mtot =
NT
X
i=1
O (DTi )rank(Ti ) = O (poly (N) poly (D)) ,
(3.7)
where D is the maximum of DTi taken over all tensors {Ti } of the
considered TN. Here we also assumed that the rank of each tensor is
bounded by a constant.
3.4 matrix product states (mps)
C
i1
i2
A1 A 2
iN
i1
i2
AN
iN
Figure 8: Matrix Product State representation of a quantum many-body
wavefunction for N particles.
3.4
matrix product states (mps)
In this subsection, we aim to give a rather panoramic view on the
probably most famous example of TN states, namely the family of
Matrix Product States (MPS). For instance, they lie at the basis of the famous Density Matrix Renormalization Group (DMRG) algorithm [51],
which has established itself as one of the most powerful numerical
techniques for simulating strongly correlated quantum systems in 1D.
For other families of TN states, like e.g. Projected Entangled Pair States
(PEPS) 2 the interested reader is referred to Ref. [28] and references
therein.
Matrix Product States are TN states that are made of tensors contracted in a pattern of a one-dimensional chain. Fig. 8 shows the representation of a quantum many-body wavefunction for N particles as
an MPS with open boundary conditions (OBC). Therefore, an MPS reproduces the one-dimensional physical geometry of the system. Every site in the many-body system has a corresponding tensor in the
MPS. The individual tensors are glued together by the contraction of
the bond indices. They can take χ different values, where χ is called
the bond dimension. The open indices correspond to the physical
degrees of freedom of the local Hilbert spaces and can take up to
d values. Matrix Product States can be easily extendend to periodic
boundary conditions (PBC), or the thermodynamic limit. In the latter case
one has to choose a fundamental unit cell that is repeated infinitelymany times. Both cases are shown in Fig. 9. Note that for fixed open
(physical) indices the corresponding coefficient is indeed represented
as a product of matrices (rank-2 tensors)3 . This explains the name
“Matrix Product States”.
Canonical forms
The representation of a quantum many-body state by a MPS is not
unique, since due to the matrix product structure we have a gauge free2 The family of PEPS is a natural generalization of MPS to higher dimensions.
3 Except for the first and the last tensor in the case of OBC, which are vectors.
19
20
tensor networks
(a)
A1 A 2
AN
i1
iN
i2
(b)
A
A
A
A
A
Figure 9: Matrix Product States of (a) a quantum many-body wavefunction
for N particles with periodic boundary conditions and (b) a 1-site
translational invariant system in the thermodynamic limit.
dom in the bonds. In other words, between any two matrices we can
insert an arbitrary invertible matrix M and its inverse without changing the state. However, there exist different choices which remove this
non-uniqueness. A particular useful choice is the so-called canonical
form [47, 48]. A given MPS with OBC and bond dimension χ is in its
canonical form if, for every bond index α, the index corresponds to
the labeling of Schmidt vectors in the Schmidt decomposition of |ψi
across that index, i.e:
|ψi =
χ
X
α
R
λα φL
α ⊗ φα .
(3.8)
R Here λα denote the Schmidt coefficients, and φL
are the
α , φα
orthornormal Schmidt vectors. In the case of a finite system with
N sites this means that we have the following decomposition of the
coefficient of the wave function:
[1]i
[1] [2]i
[2] [3]i
[3]
[N−1] [N]i
Ci1 i2 ...iN = Γα1 1 λα1 Γα1 α22 λα2 Γα2 α33 λα3 · · · λαN−1 ΓαN−1N ,
(3.9)
where the tensors Γ correspond to changes of basis between the different Schmidt basis and the computational (spin) basis, and the diagonal matrices λ contain the Schmidt coefficients. For an infinite
MPS with one-site translation invariance, the canonical form is even
simpler, since only one tensor Γ and one λ is needed to describe the
whole state. The TN diagrams for both cases are shown in Fig. 10.
There a few properties that make the canonical form very useful for
calculations. For example, it gives easy access to the eigenvalues of
the reduced density matrix of different bipartitions, which are just
the squares of the Schmidt coefficients. This is very useful if one is
interested in calculating of e.g. entanglement spectra or entanglement
entropies. Another big advantage is that calculations of expectations
values of local operators simplify a lot, see Fig. 11. The biggest advantage of the canonical form is that it provides a natural truncation
scheme for the bond indices of an MPS. At each simulation step one
only keeps the χ largest Schmidt coefficients as an approximation.
From a mathematical point this is equivalent to a well known problem, namely the “low-rank approximation” of a matrix. The physical
3.5 construction of an mps
(a)
(b)
Γ1
λ1 Γ2
λ2 Γ3
λ3 Γ4
Γ
λ
λ
λ
Γ
Γ
Γ
Figure 10: Canonical form of (a) a 4-site MPS and (b) an infinite MPS with
1-site unit cell.
intuition behind is that one reduces the rank of the matrices which
carry the quantum corrrelations, and therefore compresses the MPS
in the amount of entanglement it can support. The canonical form
and this compression in entanglement will play a crucial role in the
iTEBD algorithm presented in the next section.
Besides the canonical form presented so far, there a few related
choices of fixing the gauge degree of freedom. For example, one obtains the so called left-canonical (right-canonical) form by absorbing all
the Schmidt coefficients into the tensors to their left (right) in the
canonical form. The resulting tensors satisfy the normalization conditions depicted in Fig. 12. In practice a mixture of the the two forms
turns out to be very useful. For example, this so called mixed canonical form is crucial for improving speed and numerical stability in the
iDMRG algorithm which will be described in the next section. The
mixed canonical form is obtained by choosing a site as the orthogonality center of the MPS, and imposing that all sites to the left and
right satisfy the left-canonical and right-canonical normalization conditions, respectively. For example, one advantage is that the evaluation of single-site operators at the center site involves only the operator, the center matrix and its hermitian conjugate, see Fig. 13.
3.5
construction of an mps
In this section, we want to discuss briefly two different ways to obtain an MPS. First we show that any pure quantum many-body state
admits a MPS representation if the bond dimensions are sufficiently
large. Then we explain a hypothetical preparation from maximally
entangled states which allows us to understand why the class of
MPS is sucessfully used to simulate the low-energy states of local
one-dimensional Hamiltonians with a gap.
21
22
tensor networks
(a)
Γ1
λ1 Γ2
λ2 Γ3
λ3 Γ4
λ4 Γ5
λ2 Γ3
=
O
λ3
O
Γ1* λ1 Γ2* λ2 Γ3* λ3 Γ4* λ4 Γ5*
λ2 Γ3* λ3
Γ
λ
(b)
λ
Γ
λ
Γ
λ
Γ
λ
Γ
=
O
Γ* λ
Γ* λ
Γ* λ
Γ
Γ* λ
Γ*
λ
O
Γ* λ
λ
Figure 11: Expectation value of a single-site observable for an MPS in canonical form: (a) 5-site MPS and (b) infinite MPS with 1-site unit cell.
(a)
(b)
=
=
Figure 12: A matrix that is part of a (a) left-canoncial MPS or (b) a rightcanonical MPS, and its hermitian conjugate contract to the identity when contracted (a) over their left indices and their physical
indices or (b) over their right indices and their physical indices.
O
=
O
Figure 13: Since all tensors to the left (right) of the orthogonality center, here
depicted as a red matrix, are left- (right-) canonical, the calculation of any expectation value of a single-site operator O acting on
the center site requires only the contraction of the operator with
the center matrix and its conjugate.
3.5 construction of an mps
MPS as successive Schmidt decompositions
Let us consider a quantum many-body sytems consisting of N particles with d degrees of freedom. The coefficients of any pure quantum
many-body state describing the system in a state
|ψi =
d
X
i1 ,i2 ,...iN =1
Ci1 i2 ...iN |i1 i ⊗ |i2 i ⊗ · · · ⊗ |iN i
(3.10)
can be represented as an MPS in canonical form by making use of
successive Schmidt decompositions [49]. First we perform a Schmidt
decomposition between site 1 and the complementary N − 1 sites, and
then expand the left Schmidt vectors in terms of the original basis,
|ψi =
d
d
X
X
i1 =1 α1 =1
E
[2,...,N]
[1]i [1]
.
Γα1 1 λα1 |i1 i ⊗ φα1
(3.11)
In the above equation the matrix Γ is the corresponding change of basis matrix, and λ contains the Schmidt coefficients. We then proceed
[2,...,N]
i in terms of the baby expanding the right Schmidt vectors | φα1
sis given by the tensor product of the original basis on site 2, and the
basis for subsystem [3, . . . , N] consisting of the right Schmidt vectors
[3,...,N]
i according to a Schmidt decompostion between sites [1, 2]
| φα2
and [3, . . . , N], i.e.
d
d2
E
X
X
[2,...,N]
[2]
[3,...,N]
[2]i
i
=
Γα1 α22 λα2 |i2 i ⊗ | φα2
φα1
(3.12)
i2 =1 α2 =1
If we substitute Eq. 3.12 in Eq. 3.11, we obtain
|ψi =
d
X
d
d
X
X
2
i1 ,i2 =1 α1 =1 α2 =1
[1]i
[1] [2]i
[2]
[3,...,N]
Γα1 1 λα1 Γα1 α22 λα2 |i1 i ⊗ |i2 i ⊗ | φα2
i.
(3.13)
By iterating the above procedure for the right Schmidt vectors
[3,...,N]
[4,...,N]
[N]
| φα2
i, | φα3
i , . . . , | φαN−1 i one arrives at the canonical form
of an MPS, see Fig. 14. Note, this does not necessary mean that this
representation is efficient. In fact, per construction the range of the
bond dimension in the middle of the MPS has to be exponentially
large in the system size to be able to represent any state in the Hilbert
N
space, namely up to d 2 . However, the whole point of MPS is that
low energy states of local Hamiltonians with a gap can typically be
approximated very well by an MPS where the bond dimension is a
constant or scales at least polynomially in the system size χ [46, 24].
The bond dimension χ can be regarded as a refinement parameter.
This idea is illustrated in Fig. 15. By increasing the bond dimension
we increase the subset of the many-body Hilbert space we can faithfully represent by the MPS.
23
24
tensor networks
i1
Γ1
i2
i
iN
Schmidt decomposition
3
λ1
i1
i2
i3
iN
Γ1
λ1 Γ2
λ2 Γ3
i1
i2
i3
λN-1 ΓN
λ3 Γ4
iN
Figure 14: Exact representation of a pure quantum many-body state by
an MPS obtained by successive Schmidt decompositions and
changes of basis, see text.
Many-body Hilbert Space
N/2
χ=d
χ=1000 1D Area-law states
χ=100
χ=10
N/40
χ=d
N/20
χ=d
Figure 15: Onion-like structure of the Hilbert space: The higher the bond
dimension of an MPS, the more states are accessible in the manybody Hilbert space. For a bond dimension exponentially large in
the system size, eventually, every state can be represented as an
MPS.
3.5 construction of an mps
MPS and the Valence Bond Picture
A less technical and physically intuitive way of thinking about MPS
provides the so-called valence bond picture [45, 30]. In the valence bond
picture each of the N particles with d degrees of freedom is replaced
by a pair of virtual particles of dimensions χ. Every pair of neighboring virtual spins corresponding to different sites are assumed to be in
a maximally entangled state, which means that the state of this pair
is described by
χ
X
1
√ |αi |αi
χ
(3.14)
α=1
Then we can apply a map on each site which projects two virtual
systems with dimension χ into a single system with the real physical
dimension d,
P=
χ
d
X
X
i=1 α,β=1
[i]
Aαβ |ii hα| hβ| .
(3.15)
The state obtained in this way has exactly the form of an MPS made
of the matrices which define the projection P (see Fig. 16). Let us
mention here two nice features of the description of an MPS in terms
of maximally entangled pairs. First, in this picture it becomes particularly clear that MPS satisfy a one-dimensional area-law for the
entanglement entropy. Let us for example look at any connected partition of the system, e.g. the red sites in Fig. 16(d). The boundary
lines, represented by the dashed black lines in the figure, will always
cut exactly two of the maximally entangled states. We know that the
entanglement entropy between these virtually particles is log2 χ, and
therefore we can conclude that the entanglement entropy between the
considered subsystem and its complement is
S = 2 log2 χ = const.
(3.16)
This explains the sucess for MPS in simulation one-dimensional systems, since that is exactly the scaling behavior expected of the lowenergy states of gaped Hamiltonians in 1D. We also see that the
amount of entanglement the MPS can support is determined by the
bond dimension χ. As a second useful feature we want to mention
that the valence bond picture provides an ansatz for a natural generalization to higher dimensional physical systems. For example, the
scheme can be extended to two dimensions by replacing every physical particle with four virtualy particles. This would be the valence
bond picture of a PEPS[28].
25
26
tensor networks
(a)
(c)
↵
X 1
p |↵i |↵i
↵=1
(b)
(d)
↵
i
P =
d
X
X
i=1 ↵, =1
1 [i]
p A↵ |ii h↵| h |
Figure 16: (a) Graphical representation of a maximally entangled state. (b)
The blue circle with a outgoing leg represents the map from two
virtual systems into the real system on the site. (c) MPS in the
valence bond picture. (d) MPS satisfy a one-dimensional area law,
see text.
3.6
matrix product operators
Let us now consider an operator O acting on N sites, i.e.
d
X
O=
i1 ,i2 ,...iN ,
j1 ,j2 ,...jN =1
Wi1 i2 ...iN j1 j2 ...jN |i1 i2 . . . iN i hj1 j2 . . . jN | . (3.17)
There exist a natural generalization of the MPS formalism to an operator, namely its representation as a Matrix Product Operator (MPO).
An MPO consists of a contraction of rank-4-tensors, see Fig. 17. The
j1
j2
jN
j1
j2
jN
iN
i1
i2
iN
W
i1
i2
Figure 17: Representation of an operator as a Matrix Product operator.
application of an MPO to an MPS, gives another MPS with a bond
dimension that is the product of the original bond dimension of the
MPS and the bond dimension of the MPO. This is shown in Fig. 18.
By analogy to the construction of an MPS, one can prove that any operator can be written as an MPO using sequential SVDs. But, this conceptual way might be expontially complicated. However, many local
3.7 ground state calculations in one dimension
(b)
(a)
1..χ
1..κ
1..κχ
Figure 18: (a) Acting with an MPO on an MPS produces another MPS. (b)
The bond dimension of the new MPS is the product of the bond
dimension of the original MPS and that of the MPO.
operators have often exact representations with small bond dimension. For the explicit construction we refer the reader to the literature,
see e.g. Refs. [33, 11] and references therein.
3.7
ground state calculations in one dimension
In the following, we want to introduce two conceptual different TN
algorithms for ground state calculations of physical systems in the
thermodynamic limit, namely the Infinite Time Evolving Block Decimation (iTEBD) algorithm [48] based on imaginary time evolution, and
a variatonal ansatz called Variatonial MPS or Infinite Density Matrix
Renormalization Group (iDMRG).
3.7.1
Infinite Time Evolving Block Decimation (iTEBD)
In quantum mechanics, the time evolution of a state at an intital time
t = 0 is given by
|ψ (t)i = exp (−iHt) |ψ (0)i ,
(3.18)
if the Hamiltonian H is independent of time. Here we are interested
in the rotation to imaginary time t → −iτ, since an approximation of
the ground state |ψgs i can be found via
exp (−Hτ) |ψ0 i
,
τ→∞ kexp (−Hτ) |ψ0 ik
|ψgs i = lim
hψgs |ψ0 i 6= 0,
(3.19)
where |ψ0 i is an arbitrary initial state that has non-zero overlap with
the ground state4 . The idea of iTEBD is to implement the imaginary
(or real) time evolution on MPS5 . As a first step we discretize the time
and split the time-evolution operator U into m small imaginary time
steps δτ,
m
U (τ) = e−τH = e−Hδτ
= U (δτ)m ,
(3.20)
4 This can be easily seen by an eigenfunction expansion of |ψ0 i in terms of energy
eigenfunctions.
5 Other TN states like PEPS could also be considered.
27
28
tensor networks
where m = τ/δτ 1. For the sake of simplicity, let us assume that
the Hamiltoninan H consists of a sum of two-body nearest-neighbor
terms,
X
H=
hi,i+1 .
(3.21)
i
One can then decompose the Hamiltonian into a sum of even and
odd parts,
X
X
H=
hi,i+1 +
hi,i+1 = Heven + Hodd ,
(3.22)
i,even
i,odd
such that within both Hamiltonians all terms commute with each
other. Using a first-order Suzuki-Trotter expansion[43] we can approximate the time evolultion operator to first order in δτ by
e−Hδτ = e−(Heven +Hodd )δτ = e−Heven δτ e−Hodd δτ + O δτ2 ,
(3.23)
where
e−Heven δτ =
Y
i,even
e−Hodd δτ =
Y
i,odd
e−hi,i+1 δτ ≡ UAB ,
e−hi,i+1 δτ ≡ UBA .
(3.24)
(3.25)
Eqs. 3.23 and 3.25 show that the time evolution can be traced back to
a sequence of two-body gates, where the two-body gate gi,i+1 between
site i and i + 1 is given by
gi,i+1 = e−hi,i+1 δτ .
(3.26)
The imaginary-time evolution can finally be simulated by m 1
repetitions of the operator
Y
Y
U (δτ) =
gi,i+1
gi,i+1 = UAB UBA .
(3.27)
i,even
i,odd
The application of the operator UAB on an infinite MPS in canonical
form with a two-site translational invariance is shown as a TN diagram in Fig. 19. Since the action of the gates preserves the two-site
invariance, only the tensors Γ A , Γ B , λA , λB need to be updated. Let us
now formulate the final iTEBD algorithm for calculating the ground
state of an infinite 1D system. Starting from an initial MPS in canonical form |ψ0 i with bond dimension χ, one has to repeat the following
steps:
1. Infinitesimal evolution (even part): apply UAB on the MPS, getting
a new MPS |ψ 0 i with bond dimension χ 0 > χ.
2. Truncation: compress the MPS |ψ 0 i from bond dimension χ 0 to χ.
3.7 ground state calculations in one dimension
λB ΓA λA ΓB λB ΓA λA ΓB λB ΓA λA ΓB λB
g
g
g
~ ~ ~
~ ~ ~
~ ~ ~
λB ΓA λA ΓB λB ΓA λA ΓB λB ΓA λA ΓB λB
Figure 19: Two diagrammatical representations of UAB |ψi: (a) as two-site
gates acting on the sites and (b) as new MPS with the same invariance under shifts by sites.
3. Infinitesimal evolution (odd part): apply UBA on the MPS |ψ 0 i with
bond dimension χ, getting a new MPS |ψ 00 i with bond dimension χ 00 > χ.
4. Truncation: compress the MPS |ψ 00 i from bond dimension χ 00
to χ.
Of course, in practical applications one has to implement a termination condition, e.g. fixing the number of time steps. A detailed diagrammatic description of the steps 1 and 2 can be found in Fig. 20.
The iTEBD algorithm
requires computational space and time that
scale as O d2 χ2 and O d3 χ3 .
3.7.2
Infinite Density Matrix Renormalization Group (iDMRG)
In the following we introduce the so-called infinite variational MPS or
infinite DMRG algorithm. Instead of simulating an evolution in imaginary time as in the iTEBD algorithm, the approach here relies on the
variatonal principle. The educated guess on the trial wave function
is based on the entanglement properties of 1D quantum many-body
sytems. In particular, we want to approximate the ground state of a
Hamilitonian expressed as an MPO by minimizing
E [|ψi] =
hψ| H |ψi
hψ |ψi
(3.28)
over the family of MPS with bond dimension χ or equivalently, using
a Lagrange multiplier λ enforcing normalization, by finding
min (hψ| H |ψi − λ hψ |ψi) .
|ψi∈MPS
(3.29)
29
30
tensor networks
λB ΓA λA ΓB λB
γ
α
(i)
Θ
γ
α
j
i
X
-1
λB (λB) X
i
γ
α
i
(iv)
γ
-1
~A
λ' Y (λB) λB
[jγ]
(iii)
~A
λ' Y
α
(v)
Θ
[αi]
j
i
g
(ii)
~A
λ' Y
X
[jγ]
[αi]
j
(vi)
~ ~ ~
λB ΓA λA ΓB λB
α
i
j
γ
j
Figure 20: (i) First we contract the tensors into a single tensor Θαijγ ,
and (ii) reshape it into a matrix Θ[αi][jγ] by an index fusion
of the left (right) bond index with the left (right) physical index. (iii) Then we compute the singular value decomposition
P
A
Θ
=
, (iv) and reshape the matrices X
X
λe0 Y
[αi][jγ]
β
[αi]β
β β[jγ]
and Y into rank-3-tensors by undoing the index fusion. (v) We introduce λB back in the tensor network and (vi) form new tensors
−1
−1
A
Γ ˜A = λB
X , Γ ˜A = Y λB
. We also truncate λe0 containing
β
the χ 0 Schmidt coefficents back to bond dimension χ by keeping
the χ largest values.
3.7 ground state calculations in one dimension
(a)
γ
α
[αiγ]
i
(b)
|ψ i
H
iψ |
α
i
γ
[βjδ]
β
j
[αiγ]
δ
Figure 21: (a) Transformation of a 3-rank tensor into a vector by merging
the indices. (b) Procedure to get the effective Hamiltonian for the
third tensor in a 5-site MPS.
Finite DMRG
Before we turn to discuss the method for infinite systems, let us first
briefly consider the finite case to develop our intuition and to introduce the basic tools and notions. In this case the above minimization
is performed by adjusting all tensors in the MPS for all sites in order to make the expectation value of the energy the lowest possible.
Ideally, this is done simultaneously. However, this global optimization problem is in general quite difficult and unfeasible. Therefore,
one usually follows a sequential approach, i.e. optimizes tensor by
tensor. In practice, one picks, e.g. randomly, one tensor in the MPS
and minimizes with respect to its coefficients, while all other tensors
remain unchanged. In terms of the chosen tensor, which we call A,
the mimization problem defined by Eq. 3.29 can be written as
~ † Heff A
~ − λA
~ † NA
~ . (3.30)
min (hψ| H |ψi − λ hψ |ψi) = min A
A
A
~
In the above equation, all coefficients of A are arranged as a vector A
as shown in Fig. 21(a), Heff is an effective Hamiltonian, and N is a
normalization matrix. The effective Hamiltonian and the normalization
matrix can be considered as the enviroment of tensors A and A∗ in the
two TNs for hψ| H |ψi and hψ |ψi respectively, but written in matrix
form (see e.g. Fig. 21(b)). The minimization condition
∂ ~ †
~ − λA
~ † NA
~ =0
A Heff A
(3.31)
~†
∂A
leads to the generalized eigenvalue problem
~ = λNA.
~
Heff A
(3.32)
31
32
tensor networks
Once this optimization with respect to A is done, one proceeds by
repeating the minimization for another tensor in the MPS. In this
way, one continues sweeping through all tensors several times, until
the desired convergence in expectation values is attained. Let us remark that if we start from an MPS with open boundary conditions,
this algorithm is nothing else but the Density Matrix Renormalization Group (DMRG) algorithm in the language of TNs [51, 36]. In
the case of open boundary conditions it is also always possible to
choose an appropriate gauge for the tensors, e.g. a mixed canonical
form with A as the center site, such that N = 1. Then Eq. 3.32 reduces
to an ordinary eigenvalue problem. This is very useful for practical
implementations since it avoids stability problems due to N being illconditioned, see Ref. [28]. In what follows, we always consider MPS
with open boundary conditions in mixed canonical form.
Infinite DMRG
If we start from the very beginning with an infinite system to study
systems in the thermodynamic limit, we need to modify the above
procedure. The intuition that leads to our modifications is as follows [12]. Let us assume that we were given an infinitely large and
translationally invariant system at absolut zero temperature, i.e. in its
ground state. Then, if we were to add an additional site to the system and allow it to relax, one would expect that the new site would
change to match the rest, while the other sites in the system remain
unchanged. Or in the language of MPS, let us consider the case that
we already had an inifinite MPS with bond dimension χ which represents the ground state of our system. Then adding a site to our
system would correspond to adding another tensor in the MPS. The
relaxation process could be simulated by minimizing the energy with
respect to the new tensor in the environment given by the MPS which
approximates the ground state. We would then obtain a tensor which
looks like all of the tensors in our inifinite MPS. The idea of the algorithm is to start with a representation of the infinite systen in terms of
an approximative environment. This environment is then progressively
refined by embedding new sites, allowing the sites to relax, and then
absorbing them. Eventually this will simulate the environment experienced by a single site in the infinite system in its ground state. The
infinite-system algorithm works as follows: starting from, e.g. randomly chosen, approximative environments LH and RH representing
the left and right half, with respect to the added tensor A, of the TN
for hψ| H |ψi (see Fig. 22(a)), one has to repeat
~ corresponding to the min1. Relaxation: compute the eigenvector A
~ = λA
~ and
imal eigenvalue of the eigenvalue problem6 Heff A
6 We choose A as the center cite for the mixed canonical form of the MPS.
3.7 ground state calculations in one dimension
ungroup its index to return to its original rank-3 shape. The
effective Hamiltionian is shown in Fig. 22(b).
2. Absorption (odd step): at an odd simulation step, the optimized
tensor is contracted into the left environment LH . In detail:
a) merge the first bond index and the physical index of A to
form a matrix, and compute the singular value decompositon A = UΣV † (see Fig. 23(a)).
b) Undo the index fusion for the left index of U to get back to
a rank-3 shape (see Fig. 23(a)) and compute EH as defined
in Fig. 23(b).
c) Refine the approximation for the left environment LH by
contracting EH into it, i.e. LH := LH · EH , as shown in
Fig. 23(c).
3. Absorption (even step): at an even simulation step, the optimized
tensor is analoguesly contracted into right environment RH (see
Fig. 24). In detail:
a) merge the second bond index and the physical index of A
to form a matrix, and compute the singular value decompositon A = UΣV † .
b) Undo the index fusion for the right index of V † to get back
to a rank-3 shape and compute the analogue of the tensor
EH .
c) Refine the approximation for the right environment RH by
contracting EH into it, i.e. RH := EH · RH .
Since U and V are isometries the mixed canonical form of the MPS
is preserved at every simulation step. To check for convergence it is
useful to calculate the desired expectation value after, e.g., each first
or second simulation step. For a single-site operator acting on the
added site this is easily done as shown in Fig. 13. The main computationalcost is given by the eigenvalue problem and scales therefore as
O χ3 .
Two-site Infinite DMRG
If only a single site is added at every simulation time, the bond dimension χ of the MPS is fixed from the beginning, since it is always an
upper bound for the number of non-negative singular values7 . However, one may think of situations in which it would be advantageous
to increase the bond dimension during the calculation. This limitation
can be avoided by a slight modification of the previously introduced
7 According to the SVD theorem, the maximal number of non-zero singular values
for a m × n matrix is min (m, n). One may compare with Fig. 23(a) and Fig. 24(a) to
draw the conclusion.
33
34
tensor networks
(a)
A
A
|ψi
H
α
i
[βjδ]
RH
j
RH
A*
γ
LH
β
LH
iψ |
A*
(b)
≈
[αiγ]
Heff
δ
Figure 22: (a) Definition of the approximative environments LH and RH . (b)
Definition of the effective Hamiltonian.
(a)
A
A
γ
α
U
γ
[αi]
V†
Σ
[αi]
γ
i
(b)
U
[αi]
(c)
U
β
α
i
EH
U
=
EH
β
LH
LH
H
U*
Figure 23: Odd step: (a) SVD of the optimized tensor A. (b) Definition of EH .
(c) Refinement of the left environment.
3.7 ground state calculations in one dimension
(a)
A
γ
α
U
A
α
α
[iγ]
Σ
V†
[iγ]
i
(b)
V†
[iγ]
β
β
V†
i
†
EH
V
=
(c)
γ
EH
RH
RH
H
V†*
Figure 24: Even step: (a) SVD of the optimized tensor A. (b) Definition of
EH . (c) Refinement of the right environment.
algorithm, namely one has to add two sites at each simulation step,
see Fig. 25. The infinite DMRG algorithm is then as follows:
~ corresponding to the min1. Relaxation: compute the eigenvector Θ
~ = λΘ,
~ where
imal eigenvalue of the eigenvalue problem Heff Θ
~
the effective Hamiltionian Heff and the vector Θ are defined as
shown in Fig.25(c) and Fig.25(b), respectively.
2. Absorption: the optimized tensor is simultaneously contracted
into the left environment LH and into the right environment
RH . In detail:
a) compute the singular value decomposition Θ = UΣV † (see
Fig. 25(d))
b) Undo the index fusion for the left index of U and for the
right index of V † .
c) Compute the tensors EHL and EHR as defined in Fig. 25(e).
d) Refine the approximations for the left environment LH and
for right environment RH by the contractions LH := LH ·
EHL and RH := EHR · RH shown in Fig. 25(f).
The crucial point is that, if one adds two sites at a time, the center matrix becomes a square matrix of increased dimension md × md as can
be seen in Fig. 25 (b). This allows, in principle, for a SVD truncation
in simulation step 2.(d), see also Fig. 25(d). This is especially useful if
one tries to implement symmetries on the level of the tensors, since
one can take account of them by giving the bond index a multiple index structure. Therefore, the SVD truncation on the bond index may
change the symmetry sectors one keeps. In practice this means that
35
36
tensor networks
(a)
A
(b)
B
A
B
LH
Θ
γ
α
[jγ]
j
i
RH
[αi]
Θ
A*
(c)
α
i
[αijγ]
B*
j
γ
LH
RH
β
(d)
k
Heff
U
[jγ]
EHL
U
=
[αijγ]
δ
Θ
[αi]
(e)
l
[βklδ]
H
Σ
V†
[αi]
[jγ]
(f)
EHL
LH
LH
U*
V†
EHR
=
H
EHR
RH
RH
V†*
Figure 25: Modifications for the Two-site iDMRG algorithm.
3.7 ground state calculations in one dimension
the algorithm can readapt itself to more relevant symmetry sectors,
which have more weight in terms of Schmidt coefficients. This may
lead to an improved accuracy.
37
4
THE ISING MODEL IN A TRANSVERSE FIELD
In this chapter we use the iTEBD algorithm to study some ground
state properties of the one-dimensional Ising model in a transverse
magnetic field. It was first introduced by de Gennes in 1963 to describe the order-disorder transition in ferroelectric crystals [13], and
is presumably one of the simplest systems which exhibit a quantum
phase transition1 . To the present day many physical systems have been
found where it serves as a successfull description, see e.g. Refs. [34,
42]. Since the transverse Ising model is one of the rare cases of an exactly solvable many particle problem [32], it is also a popular benchmark model. We use it here to verify the validity of our iTEBD algorithm.
4.1
ground state properties
Let us start by writing down the Hamiltonian. It is
X
X
H = −J
σzi σzi − h
σxi ,
i
(4.1)
i
where J is a coupling coupling constant, h > 0 is the strength of the
external transverse magnetic field, and σα
i are the Pauli matrices for
the α-component of the spin at site i. The first term proportional to J
describes the nearest-neighbor interaction between the spins. Here we
focus on the case J > 0 where a ferromagnetic configuration, in which
all spins are aligned parallel, is energetically favorable. The second
term proportional to h describes an applied external magnetic field,
which disturbs the preferred ordering. Therefore, it should come as
no surprise that the nature of the ground state depends upon the
value of the dimensionless parameter λ ≡ J/h. To specify this, let us
consider two opposing limits. For h = 0 and J > 0 the ground state is
either given by
O
O
|0i =
|0ii
|1i =
|1ii ,
or
(4.2)
i
i
where |0ii and |1ii are the two possible eigenstates of the Pauli matrix
σzi with eigenvalues ±1. That is, the ground state is doubly generate
and all spins are completely ordered with respect to the z-direction.
These are both ferromagnetic states. Since the Hamiltonian in Eq. 4.1
is invariant under a Z2 -symmetry transformation, namely σz → −σz ,
1 A quantum phase transition is a phase transition at zero temperature.
39
40
the ising model in a transverse field
one may expect that the ground state is given by their symmetric
combination
1
|ψsym i = √ (|1i + |0i) .
2
(4.3)
This is, however, not the case and the system in thermodynamic limit
will “choose” one of the states in Eq. 4.2 as its ground state2 . This
phenomenon is called spontaneous symmetry breaking. Let us now consider the opposing limit, i.e. J = 0 and h > 0. In this case the unique
ground state is
O
|+ii
|+i =
(4.4)
i
where |+ii = √12 (|0ii + |1ii ) is the eigenstate of the Pauli matrix σxi
with eigenvalue 1. That means all spins point in the x-direction, but
are “disordered” in the z-direction. We see that the nature of the
ground states in the limits λ → 0 and λ → ∞ are qualitatively very
different. The exact solution of the transverse Ising model shows that
each limit corresponds to a different phase. The critical point, i.e. the
point in the parameter space which separates both phases, is found
to satisfy
J
λc =
= 1.
(4.5)
h c
In many cases different phases of a system can be characterized by a
so-called order parameter which vanishes in one phase and is different
from zero in the other phase. For the transverse Ising model an order
parameter is the magnetization in z-direction, i.e.
mz = hσz i .
(4.6)
It has a finite value in the ordered ferromagnetic phase (λ > λc ) and
is zero in the disordered paramagnetic phase (λ < λc ). The ground
state energy is
Z
J π p
E0 = −
1 + h2 − 2h cos k dk.
(4.7)
2π −π
In order to prove the validity of our algorithm we compare our results
for the ground state energy with the exact formula given in Eq. 4.7
and check for the expected behavior of the order parameter mz .
2 Of which one may be favorable due to an infinitesimal perturbation like, e.g., an
infinitesimal small external magnetic field.
4.2 results of the itebd calculations
4.2
results of the itebd calculations
In our calculations we set the coupling parameter J = 1, and compute
the ground state for different values of the external magnetic field h.
Then according to Eq. 4.5 we expect the critical point for the phase
transisiton driven by the variation of the magnetic field at hc = 1.
We start from a randomly chosen initial MPS in canonical form and
perform the iTEBD algorithm sequentially for decreasing imaginary
time steps sizes δτ ∈ {0.1, 0.05, 0.01, 0.005, 0.001, 0.0005, 0.0001}. The
time step size is decreased if all the singular values are converged
within a chosen tolerance . In particular, if we denote with ~λj the
vector containing all singular values at time step j we assume them
to be converged if
~
λj − ~λj−n < · ~1χ , n ∈ N.
(4.8)
~ λ
i
where χ is the bond dimension of the MPS, |·| denotes the componentwise taken absolut value, k·k is the Euclidean norm, and ~1χ is
the χ-dimensional vector containing only ones. For our calculations
we chose n = 10, i.e. we check convergence at every time step with
respect to the singular values ten time steps before. We set = 10−5 .
The calculated ground state energy for χ = 40 and the exact result
are shown in Fig. 26. We can see that our obtained results are in good
agreement with the exact solution. In Fig. 27, we show the absolute
error ∆E as a function of the external magnetic field. One can clearly
see that the error is of the order of the chosen accuracy . In the vicinity of the expected critical point of the external magnetic field hc = 1
the error increases. This is expected, since MPS capture the ground
states properties of gapped Hamiltonians, i.e. away from the critical
point. However, this also shows that for a large enough χ they can
possibly also be used to describe accurately one-dimensional systems
at criticality. This behavior can also be seen in Fig. 28 in which the
magnetization mz = hσz i is shown for different bond dimensions.
The calculated magnetization shows the expected behavior as an order parameter for the two phases. The value at which this sudden
change of the from a finite value to a zero value happens tends to
move closer to the critical point h = 1 with increasing bond dimension. We conclude that our results for the ground state energy and
the magnetization describe faithfully the predicted behavior from the
exact solution. This shows the validity of our algorithm.
41
42
the ising model in a transverse field
Figure 26: The exact groundstate energy and the ground state energy computed via iTEBD starting from a random MPS with bond dimension χ = 40.
Figure 27: The absolut error for the energy as a function of the magnetic
field.
4.2 results of the itebd calculations
Figure 28: The calculated magnetization mz shows the expected behavior
as an order parameter. As in the inset can be seen, the expected
jump from a finite value to zero gets closer to the critical point at
h = 1 for higher bond dimensions.
43
Part III
THE SCHWINGER MODEL
5
THE SCHWINGER MODEL
In this chapter, we briefly introduce the Schwinger model [39] and its
equivalent theory on a lattice [23], which will be the starting point of
our study with TN methods. Readers who are interested in a more
detailed discussion are referred to Ref. [10], on which this section is
mainly based on.
The massive Schwinger Model is quantum electrodynamics in two
space-time dimensions. Its Lagrangian density in the continuum reads
1
L = ψ (i∂µ γµ − m) ψ − Fµν Fµν − gψAµ γµ ψ,
4
(5.1)
where
Fµν = ∂µ Aν − ∂ν Aµ .
(5.2)
The first term is the Dirac Lagrangian density for a free fermion and
the second term corresponds to the field energy of the electric field.
The third term is the interaction term. It has the important feature that
it arises from the constraints imposed by a local gauge transformation.
That means, its shape is determined by demanding the invariance of
the Lagrangian density under the following transformation
ψ 0 = eigχ ψ,
Aµ0 = Aµ + ∂µ χ,
(5.3)
where χ is an arbitrary real function of space and time 1 , i.e. χ =
χ (x, t). The Schwinger model describes the interaction of one flavor
of fermions ψ with mass m through a U(1) gauge field A with coupling g. In (1+1)D the Lorentz indices µ, ν run from 0 to 1 and the
gamma matrices satisfy analogously to (3+1)D the Clifford algebra
{γµ , γν } = 2gµν ,
(5.4)
but due to the fact that there is no spin degree of freedom in one
spatial dimension these are 2x2 matrices. Substituting the Lagrangian
of the Schwinger model into Euler-Lagrange equations for the fields
ψ and A results in the equations of motion
γµ (i∂µ − gAµ ) ψ = 0,
(5.5)
∂µ Fµν = gjν ,
(5.6)
and
1 This is what is meant by local.
47
48
the schwinger model
where jν = ψγν ψ. The theory is quantized using canononical quantization by imposing anti-commutation relations on the fermion fields
ψ† (x, t) , ψ (x, t) = δ (x − y)
(5.7)
ψ† (x, t) , ψ† (x, t) = ψ (x, t) , ψ (x, t) = 0,
and by imposing commutation relations on the gauge fields
[E (x, t) , A1 (y, t)] = iδ (x − y) ,
(5.8)
where the electric field E is defined by
E = −F01 = F10 .
(5.9)
Using this definiton of the electric field in Eq. 5.6 we get analogues to
Maxwell’s equations in (1+1)D:
∂E
= gj0 ≡ gρ,
∂x
∂E
−
= gj1 ≡ gj.
∂t
(Gauss’ law)
(5.10)
Since there is “no space” for magnetic fields in one spatial dimension,
we only obtain the analogue of Gauss’ law and an equation which
describes the dynamics of the electric field.
5.1
the schwinger model as a lattice field theory
Starting from the Hamiltonian density H in temporal gauge, A0 = 0,
1
H = −iψγ1 (∂1 − igA1 ) ψ + mψψ + E2 ,
2
(5.11)
the model can be formulated on a spatial lattice using a Kogut-Susskind
staggered formulation [26]. The equivalent lattice Hamiltonian is
X
i X † iθn
(−1)n φ†n φn (5.12)
H =−
φn e φn+1 − h.c. + m
2a n
n
+
ag2 X 2
L .
2 n n
where a denotes the lattice spacing. In this formulation the correspondence between the fermionic lattice field φn on site n and the
continuum field ψ is

!
ψ
ψupper
upper n even
φn ↔
,
ψ=
.
(5.13)

ψlower
ψlower n odd
5.1 the schwinger model as a lattice field theory
The gauge variables θn live on the links between the sites n and n + 1,
and are connected to the vector potential via
θn = −agA1n .
(5.14)
Their conjugate variables Ln , with [θn , Lm ] = iδnm , are related to the
electric field by
gLn = En .
(5.15)
Since θn is an angular variable, Ln will have integer charge eigenvalues pn ∈ Z. Therefore, the local Hilbert space spanned by the
corresponding eigenvectors |pn i is infinite, and e±iθn are the ladder
operators
e±iθn |pn i = |pn ± 1i .
(5.16)
The lattice equivalent of Gauss’ law reads
Ln − Ln−1 = φ†n φn −
1
[1 − (−1)n ] ,
2
(5.17)
which means excitations on odd and even sites create ∓1 units of flux,
corresponding to “electron” and “positron” excitations, respectively.
Using a Jordan-Wigner transformation, φn = Πk<n iσzk σ−
n , where
1
±
x
y
σ = 2 (σ ± σ ), the fermionic degrees of freedom can be mapped
to spin-1/2 degrees of freedom
X
g
H= √
2 x
n
µX
(−1)n (σzn + (−1)n )
2 n
!
X
iθn −
+x
σ+
σn+1 + h.c. .
ne
L2n +
(5.18)
n
√
Here the parameters x ≡ 1/ g2 a2 and µ ≡ 2 xm/g have been introduced. The spins live on the sites of the lattice, with σzn |sn i =
sn |sn i, and represent “positrons” on even sites and “electrons on
odd sites. An even site with s2n = −1 corresponds to an empty
state, while s2n = 1 represents an occupied positron state, and vice
versa for the odd electron sites. In (1+1)D Gauss’ law, Ln − Ln−1 =
1/2 (σzn + (−1)n ), can be used to remove the gauge degrees of freedom [22]. The resulting Hamiltonian is then
H =x
N−2
X
−
σ+
n σn+1
n=0
"
N−2
X
+
n=0
+
+ σ−
n σn+1
N−1
µ X
[1 + (−1)n σzn ]
+
2
#
n
1X
k
z
(−1) + σk .
2
(5.19)
n=0
(5.20)
k=0
Instead of the gauge variables there is a non-local, long-range interaction term (Eq. 5.20) as opposed to Eq. 5.18.
49
50
the schwinger model
5.2
chiral symmetry and chiral condensate
In this subsection, we introduce the so-called chiral condensate which
will be the quantity we aim to study with TN methods. Without attempting to go into detail, we discuss two continuous symmetries of
the Schwinger model of which one is broken after quantization. In
this context, the chiral condensate arises as an order parameter.
The Lagrangian density of the Schwinger model is invariant under
global phase transformations of the Dirac field, i.e.
ψ 0 = eiα ψ → L 0 = L,
(5.21)
where α is a real constant. Therefore, according to Noether’s theorem
(see e.g. Ref. [31]), there is a conserved current jµ associated with
every continuous symmetry. In this case the vector current
jµ = ψγµ ψ
(5.22)
is conserved, i.e.
∂µ jµ = 0.
(5.23)
This global symmetry is known to hold at any level in fermion field
theory models, although, in principle, the vacuum state could break
the symmetry [19].
Let us now consider the case of massless (m = 0) fermions. Then
the Lagrangian of the Schwinger model has another continuous symmetry, namely the so-called chiral symmetry. That means the Lagrangian
density is invariant, if one transforms ψ into ψ 0 as
ψ 0 = eiαγ5 ψ.
(5.24)
In the above equation γ5 ≡ γ0 γ1 anti-commutes with γµ for µ = 1, 2
and α is again a real constant. For example, in the Dirac representation the gamma matrices are given by2
!
!
!
1 0
0 1
0 1
γ0 =
, γ1 =
, γ5 ≡ γ0 γ1 =
.
0 −1
−1 0
1 0
(5.25)
The associated Noether current for this symmetry is the so-called
axial-vector current jµ
5 which is given by
µ
jµ
5 = ψγ γ5 ψ.
(5.26)
While the vector current in Eq. 5.22 is conserved in the quantized
theory, the axial-vector current is not. This non-conservation of the
2 See, e.g., Ref. [19].
5.2 chiral symmetry and chiral condensate
axial-vector current is called chiral anomaly or axial anomaly. The divergence of the axial-vector current reads
g
µν
∂µ jµ
,
(5.27)
5 = 2π µν F
where µν is the Levi-Civita symbol in two dimensions, see Refs.[19,
25]. As a consequence of this chiral symmetry breaking a finite chiral
condensate Σ is allowed [19]. It is defined as the following vacuum
expectation value
Σ = ψψ .
(5.28)
In the case of the massless Schwinger model, the chiral condensate
can be computed exactly (see e.g. Ref. [35]), and is found to be
eγ
(5.29)
3 ≈ 0.159929,
2π 2
where γ is the Euler-Mascheroni constant. Therefore, the chiral condensate can be regarded as an order parameter signaling chiral symmetry breaking.
Σ0 =
Chiral Condensate on the Lattice3
In this thesis, we aim to compute the chiral condensate on an infinite
lattice for the massive Schwinger model. Written in terms of spin
operators, the chiral condensate on the lattice reads
√ X
z
x
n 1 + σn
(−1)
Σ (x) =
,
(5.30)
N n
2
where the expectation value is computed in the ground state. The
naively computed chiral condensate is known to be UV-divergent. In
particular, it diverges logarithmically in the continuum limit, a → 0. It
has been argued, that this divergence comes solely from the free (g=0)
theory. In the free case the chiral condensate on the lattice Σfree (x)
can be computed exactly
!
m
1
1
q
Σfree (x) = −
K
,
(5.31)
2
πg 1 + m2
1 + m2
g2 x
g x
where K (z) is the complete elliptic integral of the first kind [1]. This
result can be used to subtract the divergence from the computed chiral condensate in the interacting theory, and therefore to renormalize
it. In other words, we can define a so-called subtracted chiral condensate
Σsub , which allows for a continuum extrapolation, by
Σsub = Σ (x) − Σfree (x) ,
(5.32)
where Σ (x) denotes the computed chiral condensate. A detailed description of our simulation and the extrapolation procedure will be
given in the next chapter.
3 In preparation of this section we used i.a. Ref. [4].
51
6
C A L C U L AT I O N O F T H E C H I R A L C O N D E N S AT E
We have a habit in writing articles published in scientific journals to make
the work as finished as possible, to cover all the tracks, to not worry about
the blind alleys or to describe how you had the wrong idea first, and so on.
Richard P. Feynman, 1965
In this chapter, we present our study of the chiral condensate in
the Schwinger Model with TN methods. First, we discuss the naive
approach to simulate the model in the thermodynamic limit in the formulation of Eq. 5.20, and comment on a major problem we have been
faced with due to the gauge character of the theory. Then we present
an alternative ansatz which overcomes this problem by implementing the gauge symmetry on the level of MPS. Finally, we discuss the
continuum extrapolation of the computed reduced chiral condensate,
and compare our results to the literature.
6.1
gauge term vs. thermodynamic limit
Different to a lot of quantum many-body systems on which TN methods have been successfully applied, a lattice gauge theory has an essential new ingredient. In particular, for a gauge theory only gauge
invariant states are physically relevant. That means in case of the
Schwinger model that only states which obey Gauss’ law are in the
physical Hilbert space. On the level of the Hamiltionian for the lattice gauge theory this constraint leads to a non-local gauge term (see
Eq. 5.20):
" n
#
N−2
X 1X
k
z
(−1) + σk .
Hg =
(6.1)
2
n=0
k=0
Taking gauge invariance into account in this way will turn out to be
a hard problem if one aims to work in the thermodynamic limit.
First approach: Long-range Interactions with iTEBD and iDMRG
Our first approach is based on a modification of the iTEBD algorithm
which was introduced in Sec. 3.7.1. Since its original form considered
Hamiltonians with nearest-neighbor interactions, we have to change
the algorithm in such a way that it, in principle, is able to model longrange interactions. A way to realize this is to represent the imaginary
time evolution operator U (δτ) (see Eq. 3.20) in terms of an MPO.
53
54
calculation of the chiral condensate
λB ΓA λA ΓB λB ΓA λA ΓB λB ΓA λA ΓB λB
UA
UB
UA
UB
UA
UB
~B ~A ~A ~B ~B ~A ~A ~B ~B ~A ~A ~B ~B
λ Γ λ Γ λ Γ λ Γ λ Γ λ Γ λ
Figure 29: Application of U (δτ) in MPO form. The result is an MPS with a
broken canonical form.
The evolution of an imaginary time-step δτ with an MPO is shown
in Fig. 29. Due to its non-unitarity, the application of the imaginary
time operator breaks the canonical form of the inital MPS. However,
the canonical form can be recovered by using the original iTEBD algorithm in Sec. 3.7.1 to apply a large sequence of trivial two-site gates
alternatively acting on even and odd bonds1 , i.e. UAB = UBA = 1.
Once the MPS is again in canonical form, it is possible to use the SVD
truncation scheme to prevent the bond dimension to grow after every
time step. A more detailed discussion can be found in Ref. [29].
Let us recall the full Hamiltonian of the Schwinger model (Eq. 5.20):
N−2
X
X
µ N−1
−
− +
[1 + (−1)n σzn ]
σ+
σ
+
σ
σ
+
n n+1
n n+1
2
n=0
n=0
" n
#
N−2
X 1X
(−1)k + σzk .
+
2
H =x
n=0
(6.2)
(6.3)
k=0
If we write the imaginary time evolution operator in a factored form
U (δτ) = e−δτH = e−δτHl e−δτHg + O δτ2 ,
(6.4)
where Hl denotes the local terms of the Hamiltonian (Eq. 6.2), one
could naively expect that the imaginary time evolution becomes tractable.
If one is able to find an MPO representation for the evolution operator of the gauge term2 (Eq. 6.3), then the evolution of the gauge part
could be done with the algorithm above, and for the local part one
could use iTEBD in its original formulation with 2-body gates.
1 A more efficient and precise procedure is described in Ref. [29]. 2 Or at least for the gauge term, since e−δτHg = 1 − Hg + O δτ2 is a good approximation for δτ sufficiently small.
6.2 idmrg with gauge invariant mps
However, it turns out that the gauge term Hg depends explicitly
on the system size N and is site-dependent. It is not clear whether
there is a reasonable way to implement this in infinite-size methods. A
naive approach could be to identify the system size with the number
of added sites in the iDMRG algorithm presented in Sec. 3.7.2. We
experienced that in this case the algorithm does not converge.
6.2
idmrg with gauge invariant mps
Let us now present an alternative ansatz which was introduced in
Ref. [9]. In this approach one starts one step before integrating out
the gauge field degrees of freedom using the Gauss’ law constraint,
namely with the Hamiltonian in Eq. 5.18:
g
H= √
2 x
X
n
µX
(−1)n (σzn + (−1)n )
2 n
!
X
+ iθn −
+x
σn e σn+1 + h.c. .
L2n +
(6.5)
n
An obvious advantge is that this Hamiltonian is local with at most
nearest-neighbor actions, and translationally invariant under two sites.
Furthermore, it allows us to be a bit more general, since only in
(1+1)D the gauge degrees of freedom can be integrated out. This
might be quite relevant for a generalization to higher dimensions.
MPO Representation for H
In the following, we give an MPO representation of the Hamiltonian
in Eq. 6.5 to be able to run a iDMRG simulation. For now we consider
a finite lattice of N ∈ 2N sites. Further we block site n and link n into
one MPS-site, such that at every MPS-site we have a fermion and a
gauge field degree of freedom. Then the Hamiltonian can be regarded
as the sum of a 1-site operator and a 2-site operator, i.e
X
H=
hn + hn,n+1 ,
(6.6)
n
where
hn =
1 ⊗ L2n
µ
n z
(1 + (−1) σn ) ⊗ 1 ,
+
2
(6.7)
and
hn,n+1 = x
iθn
σ+
· σ−
n ⊗e
n+1 ⊗ 1 +
−
−iθn
+
σn ⊗ e
· σn+1 ⊗ 1
(6.8)
(6.9)
55
56
calculation of the chiral condensate
The first factor in the tensor product ⊗ refers to the fermion degree of
freedom, and the second factor to the gauge field degree of freedom
at the MPS site. With · we denote here the tensor product between
operators acting on different MPS-sites. The Hamiltonian can be written as an MPO with bond dimension κ = 4 where tensors are given
as in Fig. 30.
Imposing Gauge Invariance
As a last step, we want to impose gauge invariance to enforce that
our algorithm works directly within the physical subspace of the full
Hilbert space. In particular, we are only interested in states |ψi that
are gauge invariant, i.e.
Gn |ψi = 0
∀n,
(6.10)
where
Gn ≡ Ln − Ln−1 −
1 z
(σ + (−1)n ) .
2 n
(6.11)
Eq. 6.10 is nothing but the discretized version of the Gauss’ law constraint for the lattice spin-gauge system.
One possibility to impose gauge invariance would be to add a
penalty term to the Hamiltonian that prefers the correct subspace
energetically. For example, one could simulate
X
H0 = H + λ
G2n ,
(6.12)
n
instead of H, and consider the limit λ → ∞. However, in this case
gauge invariance is only approximately realized, and one hast to extrapolate in another parameter λ.
Therefore, we use here a different approach in which the Gauss’ law
constraint is directly implemented on the level of the tensors in our
MPS. Essentially, this implies that many tensor components in our
MPS ansatz have to vanish, i.e. only components compatible with the
symmetry are allowed to be different from zero.
Let us again assume that we have a finite lattice of N ∈ 2N sites.
Then a general, i.e. not necessarily gauge invariant, MPS ansatz for
the spin-gauge system has the form:
X s p s p
2N
B11 C1 1 B22 C2 2 . . . Bs12N Cp
(6.13)
2N |s1 , p1 , s2 , p2 . . . p2N i ,
sn ,pn
where the matrices [Bsnn ]αβ correspond to fermionic degrees of freen
dom, and the matrices Cp
n αβ to gauge degrees of freedom. We denote the bond dimension with χ, i.e. the bond indices take the values
α, β = 1, . . . χ.
6.2 idmrg with gauge invariant mps
(a)
1
i
i
i
1
=
1
4
=
1
i
i
4
=
1
3
i
i
4
=
4
4
j
(b)
=
j
j
3
=
j
j
j
2
2
=
j
i
i
1
i
=
4
=
2
j
j
=
j
i
3
=
j
(c)
i
=
1
i
i
=
4
j
j
=
2
j
i
=
3
j
Figure 30: MPO tensors for (a) the bulk, (b) the left boundary, (c) the right
boundary. Note, we have different tensors for even and odd sites
in the bulk due to the factor (−1)n in Eq. 6.7.
57
58
calculation of the chiral condensate
From Eq. 6.10 and Eq. 6.11 we can see that Gauss’ law is basically
a prescription how to update the electric field Ln at the right link of
site n, namely
1 z
(σ + (−1)n ) .
(6.14)
2 n
That means, if there is no charge at the site n, then Ln stays with
the value Ln−1 at the left. At the same time the electric field Ln is
increased/decreased by one unit, if there is a positron/electron3 at
site n. This “update rule" can be implemented by giving the bond
indices a multipile index structure, α → (q, αq ), and imposing the
following form on the matrices in the bulk:
n
[Bsnn ](q,αq )(r,βr ) = bsn,q
δ
,
(6.15)
n
αq ,βr q+(sn +(−1) )/2,r
Ln = Ln−1 +
pn
n
[Cp
n ](q,αq )(r,βr ) = [cn ]αq ,βr δq,pn δr,pn .
If one chooses the electric field to the left of the first lattice site to
vanish, i.e. L0 = 0, then matrices representing the boundaries are
gauge invariant if:
h
i
s1 1
B1 (q,αq )(r,βr ) = bs1,0
δ(s−1))/2,r ,
(6.16)
1,βr
p2N 2N C2N (q,αq )(r,βr ) = cp
2N αq ,1 δq,p2N .
In the above equation, the indices q and r label the electric charge
sector, and are sometimes referred to as structural or charge indices.
They label the representation of the gauge symmetry group for the
index. The indices αq and βr label the degeneracy subspace within each
charge (symmetry) sector. Every bulk or boundary tensor which is
chosen according to Eq. 6.15 or Eq. 6.16, respectively, preserves the
gauge symmetry exactly. The variational freedom lies now within the
n and cpn . The rather lengthy derivation of the result can
matrices bsn,q
n
be found in Ref. [9]. We also may refer the reader to Refs. [40, 41] for
details on symmetries in TN.
Simulation with iDMRG
We use the one-site iDMRG algortihm introduced in Sec. 3.7.2 to find
a ground state approximation in the thermodynamic limit. As in the
construction of the Hamiltionian, we again block a lattice site and
a link into one MPS-site. This leads to an MPS ansatz with a twosite unit cell due to alternating spin-gauge systems for positrons and
electrons. The initial tensors are defined according to Eq. 6.15, but are
otherwise chosen randomly within the variational degree of freedom.
To obtain a system that is invariant under translations of one site,
we also block neighboring MPS-sites corresponding to a positron and
electron spin-gauge systems together. This is illustrated in Fig. 31.
3 Recall from Sec. 5.1 that an occupied positron or electron state corresponds to sn = 1
or sn = −1, respectively, and that positrons/electrons live on even/odd sites n.
6.2 idmrg with gauge invariant mps
Gauge system:
(a)
n
n+1
Spin system:
(c)
(b)
B
C
Figure 31: (a) Infinite lattice in 1D: the spins (fermions) live on the sites and
the gauge variables on the links. (b) A lattice site and the link to
the right a represented by one MPS-site where gauge invariance
is ensured by choosing the tensors as given in Eq. 6.15.
(c) Neighboring MPS-sites corresponding to a positron and electron spin-gauge systems are blocked together to make the system
one-site translational invariant.
A list of all simulation parameters is shown in Tab. 1. We calculate
the chiral condensate for four different values of the fermion mass,
m/g = 0, 0.125, 0.25, 0.5 where in each of the cases x ∈ [10, 600]. The
parameter pmax > |pn | is a truncation parameter for the infinite local
Hilbert space of the gauge variables. Physically it corresponds to a
charge truncation. In our calculations we choose pmax = 2, i.e. we
truncate the infinite dimensional Hilbert space to five dimensions4 .
Further, we set N = 500 which corresponds to adding 1000 sites
in the physical system due to the coarse-graining. At every simulation step we compute the expectation value of the gauge operator
Gn defined in Eq. 6.11 to check the gauge invariance of the ground
state approximation at that step5 . In order to get an approximation of
the (subtracted) chiral condensate in the continuum, we have to perform different extrapolations. In the following, we show exemplary
the procedure for m/g = 0.25. Fig. 32 shows the computed chiral
√
condensate as a function of 1/ x for different bond dimensions χc
and χd , and Fig. 33 shows the subtracted chiral condensate. As expected the computed results have an dependency on the chosen bond
dimensions. The influence of the bond dimension seems to be bigger
in case of smaller lattice parameters in contrast to smaller lattice con4 Ln |pn i = pn |pn i with pn = 0, ±1, ±2.
5 Since it is not obvious whether there is a step in the iDMRG algorithm which breaks
the gauge symmetry.
59
60
calculation of the chiral condensate
Parameter
Description
χc
bond dimension of the charge index
χd
bond dimension of the degeneracy index
pmax
charge truncation
N
number of added sites
x
inverse coupling (see Sec. 5.1)
m/g
(dimensionless) fermion mass
Table 1: Simulation parameters in the one-site iDMRG algorithm.
√
Figure 32: Chiral condensate for m/g = 0.25 as a function of 1/ x for different bond dimensions.
6.2 idmrg with gauge invariant mps
√
Figure 33: Subtracted chiral condensate for m/g = 0.25 as a function of 1/ x
for different bond dimensions.
√
Figure 34: Subtracted chiral condensate for m/g = 0.25 as a function of 1/ x
for different bond dimensions. Note the cases (χc , χd ) = (3, 3)
and (χc , χd ) = (5, 3) are omitted here.
61
62
calculation of the chiral condensate
Figure 35: Computed chiral condensate for m/g = 0.25 and fit for the extrapolation in the bond dimension at x = 100.
stants, where the computed results seem to be very well converged
over the whole spectrum of the chosen bond dimensions. But, overall
one could expect also convergence for small lattice constants. In order
to get an approximation for the expected values, we perform for every
x an extrapolation in the total bond dimension, i.e. χ = χc · χd → ∞.
We find that the dependency is well described by the following fit
ansatz
c
f1 (χ) = a exp (−bχ) + d + Σ (x, χ = ∞) ,
(6.17)
χ
where Σ (x, χ = ∞) is the extrapolated value of the computed chiral
condensate for the inverse coupling x. For x = 100 or x = 500 this
is shown in Fig. 35 or in Fig. 36 respectively. Finally, we perform a
continuum extrapolation, i.e. a → 0 or equivalently x → ∞. As in
Ref. [4], we use for the subtracted chiral condensate the following fit
ansatz:
log x
1
1
f (x) = Σcont. + F √ + B √ + C ,
(6.18)
x
x
x
where Σcont. is the extrapolated contiunuum value of the subtracted
chiral condensate. As one can see in Fig. 37 this ansatz describes our
data overall very well, especially in the case of larger lattice constants,
where the influence of the bond dimension was also smaller. Therefore, we also perform a continuum extrapolation for x ∈ [10, 300]
(see Fig.38), since we expect that our algorithm describes this region
more faithfully. The obtained results for the four different fermion
masses can be found in Tab. 2. The continuum extrapolations for
m/g = 0, 0.125, 0.5 can be found in App. A.2
6.2 idmrg with gauge invariant mps
Figure 36: Computed chiral condensate for m/g = 0.25 and fit for the extrapolation in the bond dimension at x = 500.
Figure 37: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0.25 attained from all x ∈ [10, 600].
63
64
calculation of the chiral condensate
Figure 38: Continuum extrapolation of the subtracted chiral condensate for
m/g = 0.25 attained from x ∈ [10, 300].
One-site iDMRG
One-site iDMRG
x in [10, 600]
x in [10, 300]
0
0.03055
0.125
Ref. [4]
Ref. [44]
exact
0.15900
0.159930(8)
0.159928(1)
0.159929
0.09847
0.09425
0.092023(4)
0.092019(2)
-
0.25
0.07054
0.06838
0.066660(11)
0.066647(4)
-
0.5
0.04410
0.04293
0.042383(22)
0.042349 (2)
-
0.75
-
-
-
0.03062(3)
-
1
-
-
-
0.023851(8)
-
2
-
-
-
0.012463 (9)
-
m/g
Table 2: Comparison: subtracted chiral condensate in the continuum.
6.2 idmrg with gauge invariant mps
Discussion of the Results
In the following, we discuss our results for the continuum extrapolation of the chiral condensate in comparison to previous studies and
argue that they have to be regarded rather as a proof of concept than
serious precision calculations.
As one can see in Tab. 2 our results are in agreement with the
results of Ref. [4] and Ref. [44] except for the value attained from the
continuum extrapolation over x ∈ [10, 600] for m/g = 0. In the latter
case the given fit ansatz in Eq. 6.18 was not able to describe our data
faithfully in the region of smaller lattice constants (see Fig. 39 and
Fig. 40 in App. A.2).
The approach in Ref. [4] is conceptually different from ours. This
study is based on finite-size DMRG calculations and used the nonlocal Hamiltonian given in Eq. 5.20. Compared to this ansatz our chosen approach is more direct, since we work directly in the thermodynamic limit without the burden of finite-size scaling effects. Furthermore, we keep the gauge degrees of freedom and implement the
Gauss’ law constraint on the level of the MPS. This allows us to start
from the local Hamiltonian in Eq. 5.18. We think this is quite relevant
for considerations to generalize the work to higher dimensions, since
the gauge degrees of freedom can only be integrated out by Gauss’
law in (1+1)D. Due to the limited time frame of a master thesis, the
bond dimensions used in our calculations are comparatively small. In
particular, for m/g = 0.25 we used χ = 91 as the highest total bond
dimension, and in the other cases it was even smaller with χ = 63.
The extrapolations in Ref. [4] were attained from calculations up to
bond dimension χ = 140. Therefore, we expected our results not to
be as accurate.
Furthermore, we used the one-site iDMRG algorithm which has,
although being more efficient, several drawbacks compared to the
two-site iDMRG algorithm. As discussed in Sec. 3.7.2 the two-site version contains a SVD truncation in the bond dimension as opposed
to the one-site algorithm. This can be quite advantageous, since it
allows one to increase the bond dimension dynamically, i.e. as the algorithm runs. In principle, one can improve in this way the accuracy
of the algorithm on the fly, but with the huge advantage that following simulations steps can build on the work of previous iterations.
Moreover, the SVD truncation can lead to an extra accuracy when
dealing with symmetries, since the algorithm can readapt to more
relevant symmetry sectors, i.e. sectors with more weight in terms of
Schmidt coefficients6 . As a consequence of these “dynamical properties” the two-site algorithm is also more robust in the sense that is not
as likely as in the one-site case to get stuck in local minima, which is
a potential source of error in variational methods.
6 Or, equivalently, in terms of singular values.
65
66
calculation of the chiral condensate
Ref. [44] is a follow-up study of Ref. [9], where the gauge invariant MPS ansatz for the Schwinger model was introduced. Besides
working with gauge invariant MPS in the thermodynamic limit, CT
symmetry, i.e. invariance by a one-site translation and charge conjugation, was exploited. The ground state calculations were done with
the so-called time-dependent variational principle (TDVP) by doing an
imaginary time evolution [21]. In this work symmetries were treated
in a more sophisticated way by distributing variational freedoms to
different charge sectors. In contrast to that our approach was more
“pedestrian” by just choosing gauge invariant initial tensors.
In light of the points discussed above, we can conclude that our
results can be seen as a further proof of concept that the MPS ansatz is
able to describe the physical relevant states for a lattice gauge theory
even in the thermodynamic limit.
7
CONCLUSION AND OUTLOOK
The tensor network approach to lattice gauge theories was motivated
by its potential to tackle problems which are inaccessible to standard
techniques, like the presence of a large finite chemical potential and
real time evolution.
In the first part of this thesis we gave an introduction to some basic
concepts on which tensor network states and tensor network methods
rely on. Amongst others, we presented the modern view of entanglement as a resource in quantum information theory or as a property to
characterize quantum many-body states. In particular, we discussed
the so-called area laws which provide the theoretical framework for
tensor network methods.
In part ii, we gave an introduction on tensor network states with a
focus on the familiy of MPS. We presented to conceptually different
algorithms to obtain ground state approximations for infinite onedimensional systems, namely the iTEBD algorithm based on imaginary time evolution and the iDMRG algorithm as a variational method.
In a case-study we investigated ground state properties of the Ising
model in a transverse field, and used its exact solution to verify the
validity of our iTEBD algorithm.
In part iii we presented the Schwinger model and its equivalent
theory on a lattice. We discussed the chiral symmetry breaking due
to the chiral anomaly, and introduced the chiral condensate as an order parameter. Then we used an MPS ansatz to study the Schwinger
model in its formulation as a Hamiltonian lattice gauge theory in the
thermodynamic limit. We discussed how the demand for gauge invariance can lead to complications, if one takes account of it in an
unfavourable way. In particular, we argued to implement the Gauss’
law constraint rather on the level of the MPS than on the level of the
Hamiltonian, since in the latter case one has to deal with non-local
and site-dependent terms which do not allow for direct simulation
in the thermodynamic limit. Further, a detailed description of our
simulation with the one-site iDMRG algorithm and the following extrapolation procedure was given. We argued that or results as shown
in Tab. 3 can be seen as proof of concept that an MPS ansatz is able
to describe the physical relevant states for a lattice gauge theory in
(1+1)D, even with relatively simple algorithms.
The work in this thesis can be extended or generalized in manifold ways. From the view of lattice gauge theory it seems natural
to consider in a next step problems with finite chemical potential or
to investigate non-equilibrium properties. In the context of real time
67
68
conclusion and outlook
One-site iDMRG
One-site iDMRG
x in [10, 600]
x in [10, 300]
0
0.03055
0.125
Ref. [4]
Ref. [44]
exact
0.15900
0.159930(8)
0.159928(1)
0.159929
0.09847
0.09425
0.092023(4)
0.092019(2)
-
0.25
0.07054
0.06838
0.066660(11)
0.066647(4)
-
0.5
0.04410
0.04293
0.042383(22)
0.042349 (2)
-
0.75
-
-
-
0.03062(3)
-
1
-
-
-
0.023851(8)
-
2
-
-
-
0.012463 (9)
-
m/g
Table 3: Results: subtracted chiral condensate in the continuum.
evolution, one could also try to study scattering processes. Furthermore, the 2-flavour Schwinger model offers many open questions,
and would therefore serve as an interesting model. One could also
consider to study non-abelian gauge theories. However, the most interesting and presumably most challenging project would be to try to
consider QED in (2+1)D with so-called fermionic tensor networks.
We hope that in the future tensor network methods will arouse a
great deal of interest in the lattice gauge theory community, and that
they one day serve as a powerful tool in the, by modern standards,
elusive regimes of finite fermionic densities or non-equilibrium dynamics.
A
APPENDIX
a.1
the singular value decomposition
The singular value decomposition (SVD) is a matrix decomposition
that applies to any real or complex matrix. It plays a central role in
many applications, rangig from e.g. image processing to optimization
or statistics [18, 2].
Let M ∈ Cn×m with n > m be given. Then there exist unitary
matrices U ∈ Cn×n ,V ∈ Cm×m , and a diagonal matrix Σ ∈ Rn×m
such that
M = UΣV †
and
Σr 0
0
!
0
,
(A.1)
where Σr = diag(σ1 , . . . , σr ), and σ1 > σ2 > · · · > σr > 0 are the
positive singular values.
69
70
appendix
a.2
continuum extrapolations of the subtracted chiral condensate
Figure 39: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0 attained from all x ∈ [10, 600] .
A.2 continuum extrapolations of the subtracted chiral condensate
Figure 40: Continuum extrapolation of the subtracted chiral condensate for
m/g = 0 attained from x ∈ [10, 300].
Figure 41: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0.125 attained from all x ∈ [10, 600].
71
72
appendix
Figure 42: Continuum extrapolation of the subtracted chiral condensate for
m/g = 0.125 attained from x ∈ [10, 300].
Figure 43: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0.5 attained from all x ∈ [10, 600].
A.2 continuum extrapolations of the subtracted chiral condensate
Figure 44: Continuum extrapolation of the subtracted chiral condensate for
m/g = 0.5 attained from x ∈ [10, 300].
73
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EIGENSTÄNDIGKEITSERKLÄRUNG
Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig
verfasst und keine anderen als die angegebenen Hilfsmittel benutzt
habe.
Mainz, 27.10.2015
Kai Zapp