Download Geometry, Frustration, and Exotic Order in Magnetic Systems

Geometry, Frustration, and Exotic Order in Magnetic Systems
Kumar S. Raman
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
November, 2005
c Copyright 2005 by Kumar S. Raman.
All rights reserved.
Abstract
This thesis considers two topics in magnetism, the first involving classical spins and the
second quantum spins. A theme running through this work is how geometric constraints
and frustration can substantially influence the qualitative physics.
The first topic[1] is the magnetization process of spin ice. Spin ice in a magnetic field in
the [111] crystallographic direction displays two magnetization plateaux, one at saturation
and an intermediate one with finite entropy. We study the crossovers between the different
regimes from the viewpoint of (entropically) interacting defects. We develop an analytical
theory for the nearest-neighbor spin ice model, which covers most of the magnetization
curve. We find that the entropy is non-monotonic, exhibiting a giant spike between the two
plateaux. The intermediate plateau and crossover region are described by a two-dimensional
monomer-dimer model with tunable fugacities. At low fields, we develop mean-field and
renormalization group treatments for the extended “string” defects which restore threedimensionality.
The second topic[2] is the construction of a family of rotationally invariant, local, S=1/2
Klein Hamiltonians on various lattices that exhibit ground state manifolds spanned by
nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well understood quantum dimer models on the
corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily
well controlled by a new parameter which is the extent of decoration of a reference lattice.
This strategy leads to Hamiltonians that exhibit i) Z2 RVB phases in two dimensions, ii)
U (1) RVB phases with a gapless “photon” in three dimensions, and iii) a Cantor deconfined
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region in two dimensions. We also construct two models on the pyrochlore lattice, one model
exhibiting a Z2 RVB phase and the other a U (1) RVB phase. This construction provides
a proof of principle that topological phases can be realized in a local, SU(2)-invariant spin
model.
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Acknowledgements
This thesis was written under the guidance of Prof. Shivaji L. Sondhi. I began working with
Shivaji late in my third year after switching to condensed matter theory from a different
field. That I am still graduating on time at the end of my fifth year and heading to a nice
postdoc (Urbana) testifies to his great skill as an advisor. The distinction between “letting
one starve”, “catching one a fish” and “teaching one how to fish” is sometimes rather subtle.
Shivaji kept me on track with plenty of insight and help but also gave me the freedom to
work so I was never deprived of the confidence which comes from being able to solve a
problem “by myself”. I acknowledge him for this and also thank him for sharing his broad
vision of condensed matter theory with me.
Prof. Roderich Moessner (ENS, Paris) was a co-advisor on both of the topics presented
here. Nearly every aspect of this work has benefited from Roderich’s careful analysis of
technical details which range from improving the noise of Monte Carlo simulations to understanding the workings of an RG calculation. The work on spin ice was done in collaboration with Dr. Sergei Isakov (Toronto). While this thesis emphasizes my contribution to that
effort, some of Sergei’s results are also presented and acknowledged in the text. The work
on RVB phases includes a discussion of a model on the pyrochlore lattice invented by Prof.
Steve Kivelson. I would also like to acknowledge Dr. Matt Hastings (LANL) for a highly
stimulating discussion which eventually led us to invent the decoration procedure. I thank
Prof. David Huse for reading the thesis and for his support during my time at Princeton.
I have benefited from interacting with members of the condensed matter physics group.
Especially helpful were the excellent courses taught by Shivaji, and Profs. Boris Altshuler,
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Ravin Bhatt, Paul Chaikin, Duncan Haldane, and Elliott Lieb. I have enjoyed interacting
with Dr. Vadim Oganesyan (Yale) and am currently working with him on a possible extension of the spin ice RG (discussed in the text) to the problem of layered superconductors.
During my stay, I did an advanced project in the mathematical physics group with Prof.
Lieb on the problem of bosonic jellium. I thank him and also Prof. Robert Seiringer for
useful conversations on this topic. I conducted an experimental project in the group of Prof.
William Happer on the depolarization of polarized xenon gas. I would like to thank him
and members (some former) of his group: Warren Griffith, Yuan-yu Jau, Peter Ouyang,
Brian Patton, Dan Walter, and especially Nick Kuzma.
I am grateful to the physics department for providing me with a teaching assistantship
for each semester of my stay and I have benefited from interactions with many students,
faculty, and staff. In this regard, I would especially like to acknowledge Profs. David Huse,
Peter Meyers, Lyman Page, and Stew Smith, and Dr. Steve Smith. I am also grateful
for financial support which I received during my final year from the McGraw Center for
Teaching through its AI liaison program.
I would like to thank Pat Barwick, Martin Kicinski, and Laurel Lerner, for helping me
with various administrative tasks through the years.
I am fortunate that during the past five years, I have had the support of many colleagues
who are also friends. In this regard, I would like to acknowledge Toufic Suidan, Sasha
Baitine, Chris Beasley, Latham Boyle, Shoibal Chakravarty, Pedro Goldbaum, Karol Gregor, Kevin Huffenburger Subroto Mukerjee, David Olson, Vassilios Papathanakos, Srinivas
Raghu, and Emil Yuzbashan. I would like to also collectively acknowledge a large number
of friends outside of the Princeton physics department.
Finally, I turn to my family. Padma, Josh, Ravi, Jaya, and many other relatives have
helped me manage the emotional aspects of the graduate school process. The opportunity
for me to pursue a career in physics may never have arisen were it not for personal sacrifices
made by the older generation of my family long before I was born, particularly my uncles
G. Natrajan and G. Balachandran on my father’s side and my grandfather, P. V. Chandra,
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on my mother’s side. However, above all I acknowledge my parents G. S. Raman and Gita
S. Raman. Their contribution to this work is the kind of debt which can not be quantified
let alone repaid. I close by dedicating this thesis to the two of them.
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Contents
Abstract
iii
Acknowledgements
v
Contents
viii
1 Introduction
1
2 The magnetization process of spin ice in a [111] magnetic field.
6
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Model and notation
2.3
The two [111] magnetization plateaus
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
11
. . . . . . . . . . . . . . . . . . . . .
13
2.3.1
Low field termination: string defects . . . . . . . . . . . . . . . . . .
14
2.3.2
High field termination: monomer defects . . . . . . . . . . . . . . . .
16
2.3.3
Interaction of defects . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
The low field regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4.1
Mean field calculation . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4.2
RG calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4.3
Comparison with simulation . . . . . . . . . . . . . . . . . . . . . . .
23
2.5
The high field regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.6
Crossing points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.7
Relation to experiment, other theories, and applications . . . . . . . . . . .
27
2.4
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2.7.1
2.8
Cooling by adiabatic (de)magnetization . . . . . . . . . . . . . . . .
28
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3 SU(2) Invariant spin 1/2 Hamiltonians with RVB and other valence bond
phases.
38
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2
Quantum dimer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.3
Honeycomb lattice: Bipartite physics in d = 2 . . . . . . . . . . . . . . . . .
44
3.3.1
Klein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.3.2
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.3.3
Decoration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.3.4
Square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Other Valence Bond Phases in d = 2 and d = 3 . . . . . . . . . . . . . . . .
54
3.4.1
Non-bipartite lattices in d = 2
. . . . . . . . . . . . . . . . . . . . .
54
3.4.2
Non-bipartite lattices in d = 3
. . . . . . . . . . . . . . . . . . . . .
56
3.4.3
Bipartite lattices in d = 3 . . . . . . . . . . . . . . . . . . . . . . . .
56
Dynamical selection of gauge structures: pyrochlore lattice . . . . . . . . . .
57
3.5.1
The Klein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.5.2
The Kivelson-Klein model . . . . . . . . . . . . . . . . . . . . . . . .
58
Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.4
3.5
3.6
A An overview of height representation theory
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A.1 The height representation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
A.2 Application to spin ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
A.2.1 Classical dimers on the honeycomb lattice . . . . . . . . . . . . . . .
71
A.2.2 Interaction between defects in spin ice . . . . . . . . . . . . . . . . .
75
A.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
A.3 Application to quantum dimer models . . . . . . . . . . . . . . . . . . . . .
77
A.3.1 Quantum dimers on bipartite lattices . . . . . . . . . . . . . . . . . .
77
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A.3.2 Interaction of defects . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
A.3.3 Stability of the RK point . . . . . . . . . . . . . . . . . . . . . . . .
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B Mean field theory for string defects
82
B.1 Mean field calculation of the system response . . . . . . . . . . . . . . . . .
82
B.2 Correlation lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
C Renormalization group treatment of string defects
87
D Sign conventions in the overlap matrix
91
D.1 Overlaps in the fermionic convention . . . . . . . . . . . . . . . . . . . . . .
92
D.2 Honeycomb and diamond lattices . . . . . . . . . . . . . . . . . . . . . . . .
92
D.3 Other bipartite lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
D.4 Kivelson-Klein model on pyrochlore lattice . . . . . . . . . . . . . . . . . . .
93
E Spinon gap for the decorated honeycomb lattice
95
F Classical dimers on the pentagonal lattice
104
References
107
x
Chapter 1
Introduction
In conventional textbook examples of interacting many body systems, the qualitative physics,
such as the phases the system can exhibit, may be obtained from very general features about
the system geometry (for example, whether it is periodic), interaction (for example, whether
it is short-ranged or long-ranged) and symmetries. In contrast, the central theme of this
thesis is the influence of frustration, arising from microscopic details of the interplay of
interactions and geometric constraints, on the macroscopic physics. The two topics considered in this thesis are model spin systems, one involving classical spins and other quantum
spins, where frustration gives rise to exotic phase diagrams not easily described in the usual
framework of local order parameters and symmetries.
A canonical example of frustration and its consequences is the large ground state degeneracy of the classical Ising antiferromagnet on the triangular lattice. The Ising interaction
prefers neighboring spins to be oppositely aligned. If we consider a triangular plaquette and
anti-align two of the spins, then the third spin is frustrated in that whichever way it points,
it is unable to simultaneously satisfy all of its interactions. In contrast, the same interaction on a square lattice can be fully satisfied at every site via the Neel configuration. This
comparison is shown in Fig. 1.1. In the triangular case, any spin configuration where every
triangle has at least one up spin and one down spin is a ground state. The ground state
manifold is highly degenerate, the number of states increasing exponentially with system
1
2
size, while the same interaction on the square lattice has only two ground states.
?
Figure 1.1: Classical Ising spins with nearest-neighbor antiferromagnetic interaction. On
the square lattice, the interaction is optimally satisfied by the Neel configuration drawn
above. In contrast, spins on the triangular lattice are frustrated in that the third spin
is unable to simultaneously satisfy its interaction with its up and down neighbors. This
scenario does not arise on the square lattice because in that geometry two neighboring
spins do not have a common neighbor.
The system can move in this highly degenerate manifold by local spin flips and at low,
but nonzero, temperatures, the system will be described by ensemble averages over this
manifold. In contrast, a macroscopic perturbation is required to move between the two
ground states in the square lattice case. Related to this is the fact that the state of the
square lattice Ising antiferromagnet can be described by a local order parameter, for example the magnetization at a given site. Such an order parameter will be zero, upon ensemble
averaging, in the triangular case but the ground state is not “disordered” in the sense of a
paramagnet, though it shares the same macroscopic symmetries. In the paramagnetic case,
interactions are negligible compared to thermal fluctuations and each spin is essentially independent of the others. In the ground state of the triangular antiferromagnet, interactions
are strong and flipping a spin will generally require flipping neighboring spins in order to
maintain the “one up and one down per triangle” constraint. Recent studies [26, 25, 50, 22],
building on the work of Blote et. al. [24], have made important progress in characterizing
the order within such “disordered” systems using height representation theory. One feature
of the height representation is that excitations of the system appear as vortices in a height
field which is convenient for analytical treatments.
3
The topic of Chapter 2 is spin ice, where a geometrically constrained ferromagnetic
interaction gives rise to frustration. As will be discussed, experimental signatures of the
frustration include the retention of entropy at very low temperatures (when naively we
expect it should tend to zero); the failure to develop long range magnetic order despite a
ferromagnetic Curie constant; and the appearance of two plateaux in the magnetization
when a field is applied along a particular crystallographic direction. The height representation will be used to characterize the lower plateau and to analyize the string-like excitations
which cause its low-field termination.
An exponentially degenerate ground state implies a finite entropy (per spin) at zero
temperature. Assuming the third law of thermodynamics is correct, behavior such as that
described below can not literally occur in a physical system. However, frustration can give
rise to a large number of low lying states very close in energy. When viewed at energy
scales (i.e. temperatures) much larger than the characteristic level spacing, the behavior is
effectively an ensemble average over all of these states. In spin ice, the bandwidth of these
states is believed to be much smaller than experimentally relevant temperatures so that
while the physical system probably has a true ground state, it is dynamically irrelevant.
In the triangular antiferromagnet example and also spin ice, the apparent lack of an
order parameter is due to the ensemble averaging which occurs at temperatures of interest.
However, frustration can also influence the zero temperature characteristics as in the topic
presented in Chapter 3 of the thesis. There we construct SU(2) invariant spin systems that
realize the phase diagrams of quantum dimer models. The construction involves perturbing
a class of models called Klein models. These models are antiferromagnetic in nature but
also include additional terms which frustrate the system into forming singlets between
neighboring spins. The phase diagrams of these models differ substantially if the lattice is
bipartite or non-bipartite. In the case of the non-bipartite triangular lattice, the ground
state phase diagram features an RVB (resonating valence bond) spin liquid phase. A valence
bond state is a wavefunction where each spin forms a singlet pair with one of its nearest
neighbors. An RVB state is a superposition over all singlet configurations connected by
4
local resonance moves.
Spin liquids are characterized by rapidly decaying correlations, translational and rotational invariance, and the lack of a local order parameter. However, as in the classical case
discussed above, the state is not “disordered” in the paramagnetic sense either. It turns out
that while spin liquids do not have a local order parameter, they do possess a global type
of order called topological order. Fig. 1.2 explains this notion in more detail. A central
feature of topological phases is that they admit fractionalized excitations. In the RVB example, a natural excitation is the spinon, which is a spin 1/2 excitation formed by breaking
a singlet (valence bond). The name “fractionalized” arises because when a ground state is
described by spin 0 objects, the naively expected spin excitations will have integer spins
but in this example, valence bonds (spin 0 objects) admit excitations with half integer spins
(spinons). Currently, there is no definitive experimental realization of spin liquid physics
though, as will be discussed, these ideas form the basis of theoretical descriptions of a variety
of phenomena in correlated electron systems including high Tc superconductivity. [41]
The notions discussed in this brief overview will be made more precise in the respective
chapters of this work. The purpose of this introduction was to highlight the common thread
connecting the two rather different topics discussed in this thesis, namely how frustration
can give rise to exotic behavior not easily categorized by conventional paradigms.
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C1
C1
Figure 1.2: A valence bond state is a state where each spin forms a singlet pair with one of
its neighbors. The RVB spin liquid is a quantum superposition of all valence bond states
connected by local resonance moves. The above example shows a valence bond covering for
a part of a triangular lattice and the dotted lines depict the simplest local move. C1 is a
line extending through the system and we see that initially there are three bonds crossing
the line and after the flip, there is one. By inspection, we see that if the number of bonds
crossing the line is odd (or even) then this property will not be affected by local resonance
moves. We could also have drawn a horizontal line and the torus depicts the fact that there
are four distinct topological sectors (the number of bonds crossing a horizontal/vertical
line may be odd/odd, odd/even, even/odd, or odd/odd). The RVB spin liquid state is a
superposition of all valence bond coverings in a given topological sector so the state may
be labelled by its winding number. This global property is called topological order.
Chapter 2
The magnetization process of spin
ice in a [111] magnetic field.
2.1
Introduction
The name “spin ice” refers to a class of magnetic compounds that may be described by spins
on a lattice obeying a local “ice-rule” constraint. Specific examples of spin ice compounds
we will be interested in are Ho2 Ti2 O7 and Dy2 Ti2 O7 . For a review on spin ice, see Ref. [3].
The dynamical objects in models of these compounds are the large spins of the rare-earth
ions (e.g. JHo = 8, JDy = 15/2) which reside on the sites of a pyrochlore lattice, shown
in Fig. 2.1. As the figure shows, each pyrochlore site is a corner shared by two tetrahedra.
Fig. 2.2 shows a single tetrahedron inscribed in a cube with some important crystallographic
directions labelled.
An important effect of the neighboring Ti and O atoms is to cause a crystal field
anisotropy which strongly favors maximizing the component of the Ho/Dy spin pointing
along its “easy-axis” which is the local [111] direction. As Figs. 2.1 and 2.2 show, this
axis is the line joining the centers of the two tetrahedra sharing the corner where the spin
resides. In this work, we take the anisotropy energy to be infinite so that with respect to
one of its tetrahedra, the spin either points “in” towards the center of the tetrahedron or
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Figure 2.1: The pyrochlore lattice of corner-sharing tetrahedra.
“out” away from the center. The configuration space becomes more constrained when the
interaction between spins is considered. As discussed below, the net interaction may be
modelled by a ferromagnetic coupling between nearest neighbor spins. With reference to
a single tetrahedron, this implies that if a particular spin points “in”, then its interaction
is optimized if the other spins of the tetrahedron point “out”. Of course, this is also true
for the other spins so that the ferromagnetic nearest-neighbor coupling, combined with the
pyrochlore geometry and easy-axis constraint, gives rise to frustration. An optimal configuration for a given tetrahedron is one where two of its spins point inwards and two outwards;
for each tetrahedron, there are six such configurations. An optimal configuration for the
whole system is one where the spins obey what is called an “ice-rule”: on every tetrahedron,
two spins point in and two out. As is commonly the case in frustrated systems, the number
of such optimal configurations grows exponentially with system size.
8
Figure 2.2: A single tetrahedron inscribed in a cube. The edges of the cube denote the
[100] direction and the edges of the tetrahedra are the [110] direction. The [111] directions
of the pyrochlore lattice are the body diagonals of the cube, denoted by d̂κ , and indicated
by the dashed lines.
The idea of geometric frustration being the source of a large “ground state” degeneracy
was first proposed by Pauling in the context of cubic water ice Ih .[5] The mystery of water
ice was that integrating low temperature measurements of its specific heat suggests that
the entropy approaches a (non-zero) constant at zero temperature, in apparent violation
of the third law of thermodynamics. An important step in the resolution of this mystery
was the formulation by Bernal and Fowler[4] of the ice rules, based on measurements of the
crystallographic structure and bond lengths. Referring to the pyrochlore structure discussed
above, in water ice the oxygen atoms lie at the centers of the tetrahedra and are surrounded
by four hydrogens which lie on the corners of the tetrahedra, each H being “shared” by two
O’s. Bernal and Fowler proposed that a hydrogen would not be exactly on a corner but
closer to one of the two oxygens sharing it and the optimal configurations would be where
each central oxygen has two hydrogens close to it and two farther away. The resulting
degenerate manifold is exactly the ice-rule manifold discussed above for spin ice, where the
4 spins on a tetrahedron play the role of the 4 O-H bond lengths. Pauling[5] showed that
the degeneracy gives rise to a macroscopic ground state entropy which agreed well with
9
experiments.
Pauling proposed that while the true ground state of ice may be unique, thus satisfying the third law, the frustration gives rise to a large number of closely spaced low lying
states. Typical experimental temperatures are much larger than the bandwidth of these
states so that the system is effectively described by ensemble averaging over these states.
Thus, the true ground state is dynamically irrelevant and the system displays a “zero-point
entropy”. Anderson [6] suggested that Pauling’s proposal may have a magnetic analogy
and theoretically predicted spin ice roughly forty years before its first experimental realization. The discovery[9] was made when it was observed that the pyrochlore compound
Dy2 Ti2 O7 did not magnetically order at low temperatures even though measurements of
its dc susceptibility suggested a ferromagnetic Curie-Weiss constant. It was suggested that
the lack of local order was due to the system fluctuating between degenerate ice-rule states
and the hypothesis was confirmed when its measured zero-point entropy was in agreement
with Pauling’s prediction for water ice.
Spin ice is easier to work with (experimentally) than water ice and may be used to explore various properties of (cubic) water ice. Spin ice is susceptible to a magnetic field which
allows for the exploration of features without previously observed water ice analogues[7].
The highly anisotropic magnetic response of a single spin ice crystal to magnetic fields along
different crystallographic directions was first investigated numerically[14] and as unicrystalline samples became available, experimentally[10, 11, 12, 13]. However, vestiges of some
of these anisotropic features, such as a sharp spike in the specific heat for fields in the [100]
direction, were possibly seen earlier in orientationally averaged powder samples.[9]
The main subject of the present work is the magnetization process of a unicrystalline
sample of spin ice that is placed in an external magnetic field in the [111] direction. A sketch
of the exotic thermodynamic properties of spin ice in a [111] field is given in Fig. 2.3. The
striking features are the two magnetization plateaus and the entropy peak which occurs in
the vicinity of their crossover. The higher plateau corresponds to saturation: the magnetization fraction of 0.5 is the highest value consistent with the easy axis constraint. The lower
10
plateau, first predicted theoretically[14] and explored in Monte Carlo simulations,[14, 16]
has been particularly remarkable as it was found to retain a fraction of the zero-field spin
ice entropy.[11, 21, 22] Vestiges of a [111] entropy peak were seen in experiment prior to
the present study. [11] Magnetocaloric measurements of the entropy[15], published after the
present work[1] show a clear entropy peak, though somewhat smaller in magnitude than
our prediction.
The spin-ice results presented in this thesis builds on work initiated in Ref. [22] where
the thermodynamics and correlations of the [111] plateau were systematically explored.
Different regimes of the magnetization curves were identified, including the mechanisms
which terminate the lower plateau at its high and low field ends. On the plateau itself, the
system is well described by a two-dimensional antiferromagnetic Ising model on a kagome
lattice in a longitudinal field, which is in turn equivalent to a hexagonal lattice dimer
model.[21, 22, 23]. At the high-field end, the crossover between the plateaus occurs via the
proliferation of monomer defects in the underlying dimer model. At low fields, a more exotic
extended string defect restores three dimensionality. The asymptotic densities of both kinds
of defects were estimated in Ref. [22]. The various regimes are indicated in Fig. 2.3.
In Section 2.2, we develop a theoretical model for spin-ice in a [111] field. In Section 2.3,
we discuss the physics of the [111] plateau and introduce the mechanisms which terminate
it at low and high fields.
In Section 2.4, we consider the low field end of the plateau in detail. We develop mean
field and renormalization group treatments for the extended string defects, which we use
to analyze the in-plane and out-of-plane correlations. We compare these with Monte Carlo
simulations by Isakov[34], who used an efficient cluster algorithm to obtain accurate data
from the zero field to the beginning of the [111] plateau. We find that the mean field
treatment is accurate at the lowest fields, where the string density would be relatively high.
The renormalization group treatment compares well with simulation in the dilute string
limit. At even higher fields, the plateau is approached and the suppression of the entropic
activation of strings becomes apparent as a finite-size effect.
11
In Section 2.5, we study the high field end of the plateau. We present the results of
Monte Carlo simulations of a two-dimensional Ising antiferromagnet on a kagome lattice,
which we compare with an analytical treatment by Isakov[34]. We observe a giant peak in
the entropy, which even exceeds the zero field Pauling value, despite the fact that a quarter
of all spins are pinned. We show that the entropy peak is due to the crossing of an extensive
number of energy levels which have macroscopic entropies.
We conclude by comparing with experiment and discussing some potential practical
implications of the work.
2.2
Model and notation
A starting point for a theoretical description of spin ice is the following Hamiltonian, where
the spins are treated classically:
H =
Jij Si · Sj + D
(i,j)
− E
d̂κ(i) · Si
i
2
3(Si · rij )(Sj · rij ) − Si · Sj
(i,j)
+µ
5
rij
B · Si ,
(2.1)
i
The sums on (i, j) are over all pairs of spins; the spins {Si } are unit-length. The first term
is an exchange interaction where the constants Jij are expected to decay rapidly with the
distance between sites. The second term is a dipolar interaction of strength D, where rij
is the vector distance between sites and rij = |rij |. The constant D =
µ0 2
4π µ ,
where µ is the
magnetic moment (for Dy, µ ≈ 10µB ). The third term is the easy-axis anisotropy which
favors the spin pointing along the local [111] direction, denoted by d̂κ(i) . The energy scale
of the anisotropy is given by the constant E > 0,|E| ∼ 50K, which is much larger than
typical exchange and dipolar interaction strengths. We take E to be infinite, which turns
the anisotropy into a strict constraint. The fourth term is the influence of a magnetic field.
Note that the only free parameters in this theory are the exchange constants Jij .
Siddharthan et. al. [16] performed Monte Carlo simulations of Eq. 2.1 assuming a purely
dipolar interaction, truncated at the sixth neighbor piece, and found good agreement with
experiment when the antiferromagnetic exchange interaction was accounted for by reducing
12
the value of D by 25% from the above value. The nearest neighbor piece of the dipolar
interaction favors alignment of neighboring spins which, when combined with the easy-axis
constraint, gives rise to the ice rule behavior discussed above and seen in experiments. The
ice rule energy scale is somewhat reduced by the exchange interaction. It is not obvious why
a long range dipolar interaction, which is what was simulated in [16], should give behavior
consistent with keeping only the nearest-neighbor piece. That the local Ising axis varies
from site to site appears to cause a subtle cancellation of the longer ranged dipolar terms.
Siddharthan et. al. [16] truncated the dipolar interaction at the sixth neighbor piece,
after observing that the remaining terms were small corrections. Gingras et. al. [17, 18]
handled the dipolar lattice sum using an Ewald summation method which permits one to
treat the interaction to infinite range. They found the surprising feature that the infinite
distance result is closer to the nearest neighbor result than an interaction truncated at, for
example, 10 nearest neighbors. The analytical explanation for this remarkable cancellation
was provided recently by Isakov et. al. [19, 20], who demonstrated that the long-range
dipolar interaction could be replaced with a “model” interaction that has the same ground
state manifold as the nearest-neighbor piece but differs from the physical interaction by
small terms which decay faster than the 1/r 3 dipolar decay.
This recent work provides a theoretical justification for our starting point of modelling
the effective spin-spin interaction as a ferromagnetic coupling between nearest-neighbors.
A subtlety of making this approximation is that the correspondence of the (model) dipolar
ground state manifold with the nearest-neighbor ground state (ice rule) manifold does not
extend to the excited states of the two Hamiltonians. In particular, if an external field
is strong enough to take the physical system out of its zero-field ground state manifold,
then the new state need not correspond to what is obtained by similarly perturbing the
nearest-neighbor Hamiltonian. We will return to this point in Section 2.7 but for now, we
invoke the nearest-neighbor approximation which, it turns out, can explain many of the
experimental features.
Because we take the easy-axis anisotropy as infinite, it is convenient to describe the
13
system by the Ising pseudospins σi , where Si = σi d̂κ(i) . The pseudospin σi =+1(-1) if the
physical spin points into (out of) its associated up-pointing tetrahedron (“up-pointing” and
“down-pointing” are with respect to the [111] direction indicated in Fig. 2.4). We may write
an effective Hamiltonian for the pseudospins:
H = Jeff
<ij>
σi σj − gµB J
B · d̂κ(i) σi ,
(2.2)
i
where Jeff ∼ 1K[11]. Note that the ferromagnetic spin-spin interaction gives rise to an
antiferromagnetic interaction between pseudospins. The ice rule constraint says that the
pseudospins on each tetrahedron must sum to zero, |
κ σκ |
= 0. At zero temperature and
zero magnetic field, the ice rule states span the ground state manifold of our model.
2.3
The two [111] magnetization plateaus
At low magnetic fields (and low temperatures), the system will continue to obey the ice
rule, though the magnetic field will favor certain states among those in the zero-field ground
state manifold. For the calculations we will describe, it is useful to visualize the pyrochlore
lattice along one of its [111] directions as shown in Fig. 2.4. The lattice is seen to be a stack
of alternating kagome and triangular planes, the [111] direction being the direction in which
the planes are stacked. In this picture, each spin lies on a corner shared by an up-pointing
and down-pointing tetrahedron.
If a magnetic field is applied along one of the [111] directions (see Fig. 2.4), then ice
rule states where spins are aligned with the field (as much as possible, given the easy-axis
constraint) will be favored. If the [111] field is large enough, the spins in the triangular
planes, having easy-axes permitting full alignment, will completely polarize and the kagome
planes will decouple. The triangular planes are formed by the apical spins of the uptetrahedra. With respect to their up-tetrahedra, these spins are pointing out so the ice rule
requires two of the remaining spins to point in and one to point out. The result is that the
system is well-described by a two-dimensional model with its own ice-rule: pseudospins on
a kagome lattice where on each triangle, two pseudospins point up and the remaining one
14
points down. This describes the basic physics of the lower [111] plateau and the degeneracy
of the “kagome ice” manifold accounts for the finite entropy on the plateau. The higher
[111] plateau is a zero entropy state where the spins are completely aligned with the field
(as much as possible, given the easy-axis constraint); note that this state is outside of the
ice rule manifold. The crossover between the plateaus is a competition between satisfying
the ice rule and optimizing the field energy.
We now turn to a brief description of the mechanisms which terminate the lower [111]
plateau at low and high fields.
2.3.1
Low field termination: string defects
At fields slightly lower than the plateau, excitations labelled string defects[22] restore threedimensionality and are responsible for the low field termination of the plateau.
To describe these defects, we consider the entropic benefit of relaxing the condition that
the triangular planes are polarized. Suppose we flip a spin in some triangular layer. This,
by itself, would violate the ice rule for the up and down tetrahedron sharing that spin. To
restore the ice rule on the two tetrahedra, we need to also flip spins in the kagome layers
of each tetrahedron. Flipping these kagome spins requires flipping spins in each of the two
neighboring triangular layers, which requires flipping spins in the two next-nearest kagome
layers and so on. The resulting “string defect”, illustrated in Fig. 2.5, is an excitation that
extends through the system.
That such a defect must be infinite in length follows more directly from an argument
given in Ref. [22]. Each spin in the kagome layer is a member of an up-triangle which is
the base of an up tetrahedron, whose apex is part of the above triangular layer. The ice
rule implies that sum of the pseudospins on each tetrahedron is zero. These two facts imply
that the pseudospin magnetization of a [111] kagome plane is the equal and opposite of
the triangular plane above it. The triangular spins are also members of down tetrahedra
corresponding to the down triangles in the above kagome layer. Therefore, we find that
the local ice rule implies the global property that all of the [111] kagome planes have the
15
same (pseudospin) magnetization, which is equal and opposite to that of the [111] triangular
planes. On the plateau, the triangular magnetization is saturated. The preceding global
property implies that to reduce this magnetization by flipping a spin in one layer necessarily
requires simultaneously flipping spins in all of the other layers.
Despite its name, which is historical, a string defect is not a “string”, i. e. a single
extended object. Creating a string defect actually involves creating a pair of defects in each
kagome layer. A “positive” defect connects the kagome plane to the kagome plane directly
above it via a flipped spin in the intermediate triangular plane. Similarly, a “negative”
defect connects the kagome plane to the kagome plane directly below it. Two defects in a
given plane may be separated by flipping pairs of spins pointing in different directions on
neighboring triangles of the kagome plane. Every “negative” defect must be directly above
a “positive” defect in the layer below. Therefore, creating a string defect actually means
creating a defect in each kagome-triangle bilayer. In particular, a dilute gas of N string
defects merely means having N defects in each kagome-triangle bilayer (or equivalently, N
positive and N negative defects in each kagome layer with appropriate interlayer coupling)
and in no sense are particular defects associated with a particular “string”.
We may estimate the condition for having a string defect. The energy cost of a string
defect may be found by noting that in each kagome-triangle bilayer, we need to flip one
triangular and one kagome spin that were previously aligned with the field. The cost per
bilayer is Es = 8gµB JB/3 so the total energy of a string E ∼ BL, where L is the system
size. The thermodynamic state containing a single string defect has an entropy due to our
choice of which spin to flip in each triangular layer. This suggests that the entropy of the
system S ∼ L ln L. The free energy balance implies that for nonzero temperatures and a
given magnetic field, string defects are favored in a sufficiently large system. For a given
system size, strings are favored at sufficiently low magnetic fields.
16
2.3.2
High field termination: monomer defects
On the plateau, the magnetization of the triangular sublattice is saturated and we may
consider each kagome plane separately. Thus, the 3-dimensional model may be mapped
onto a 2-dimensional one. Each triangle on the kagome plane contains two up pseudospins
(σ = 1) and one down pseudospin (σ = −1). The states satisfying this constraint are in
one-to-one correspondence with dimer coverings of the dual honeycomb lattice as sketched
in Fig. 2.6.[21, 22, 23] The entropy on the plateau may be obtained from the exact solution
of the honeycomb dimer problem. [21, 22] In this manner, it is found that the plateau retains
an extensive ground state entropy, S/kB = 0.080765 (per spin).
If we flip a down (pseudo)spin it violates the ice rule on two triangles, one being uppointing and the other down-pointing as may be readily seen by inspecting Fig. 2.6. Analogous to the discussion of strings, these triangles correspond to positive and negative defects
which can move on the lattice. In terms of the dimer description, creating a pair of defects
involves replacing a dimer with two monomers (in the dimer description of a string, the pair
of defects created in each kagome plane have two dimers/site). The energy cost for creating
two monomers is 2E = 4Jeff − 2gµB JB/3. This is balanced against the entropic benefit
in that we can place the monomers anywhere on the lattice. The energy cost vanishes at
a critical field Bc = 6Jeff /(gµB J). At higher fields, the monomers proliferate leading to
complete saturation and an ordered state with zero entropy.
2.3.3
Interaction of defects
The picture of string and monomer defects presented above is a simplification in that the
defects interact. The interaction is entropic in origin and stems from the fact that if the
defects are held fixed at various positions, there are a number of ways in which the remaining
sites may be covered by hard-core (one dimer/site) dimers. Defect configurations permitting
many dimer coverings would appear more often in a partition function sum than more rigid
configurations. In this sense, we may assign an entropy to each configuration of defects and
if these defects are dilute, we may represent the entropy of a configuration as the sum of
17
pairwise interaction terms.
In Appendix A, it is shown that at long wavelengths the entropy of two defects in the
same plane as a function of defect separation is:
S(|r1 − r2 |) = −κp1 p2 ln
|r − r | 1
2
τ
(2.3)
where κ = 12 ; (ri , pi ) are the position and charge (±1) of the ith defect; and τ is a short
distance cutoff of order the lattice spacing. Of course, the overall entropy can not actually
be negative so more precisely, this function is the entropy relative to that of the reference
state of having no defects (which is essentially the case where the two defects are infinitely
far apart). This amounts to an overall constant, which we have dropped, of order ln(L/τ ),
L being the system size, which arises because the number of dimer coverings increases
exponentially with the system size. This coarse-grained description is expected to work
at long wavelengths but the details of the kagome lattice enter through the value of κ, as
discussed in Appendix A. Eq. 2.3 is expected to capture the leading in-plane interaction
at long wavelengths for both the monomer and string defects (in the latter case, we are
referring specifically to the local defects created in the planes). This is because both types
of defects (in fact, any kind of local defect) appear as vortices in an underlying height field,
as discussed in Appendix A. In order to maximize its in-plane entropy, the system will
prefer defects of the same (opposite) sign to be far apart (close together).
Since the form of this interaction is that of 2d electrostatics, we will refer to the defects
as “charges” but this should not conjure up images of wires or vortex lines. Eq. 2.3 only
gives the interaction between two defects in the same plane. In the monomer case, this is
the only interaction since the individual planes are decoupled. In the case of strings, we
have the additional constraint that a positive charge in layer k is rigidly connected to a
negative charge in layer k + 1. This negative charge can interact with the other charges in
the k + 1 plane so the result is an interlayer coupling between defects in nearest neighbor
planes. The details of this coupling are developed in the calculations described in the next
section.
18
2.4
The low field regime
We now consider the regime of the magnetization curve (Fig. 2.3) near the low field termination of the plateau. String defects will be present by the arguments discussed above. In
the next two subsections, we investigate the in-plane and out-of-plane correlations for these
defects when the density of these defects is in the medium (mean-field) and low (renormalization group)density regimes. We obtain the dependence of the defect density on temperature
and field in both of these limits and compare the analytic results with numerical simulation
of the magnetization.
2.4.1
Mean field calculation
If the number of string defects is fairly large, we may expect the interaction to be sufficiently
screened to justify the use of variational mean field theory[32]. We consider a layered system
of two-dimensional planes (indexed by the label k which ranges from −K to K) where each
plane contains N positive and N negative charges that interact logarithmically. The string
constraint requires that each positive charge in layer k is rigidly connected to a negative
charge in the layer k + 1. We impose a periodic boundary condition to connect the positive
charges in the Kth layer to the negative charges in the −Kth layer.
We formally impose the constraint by writing the “Hamiltonian” in terms of positive
charges alone. The planes are stacked in the z-direction. Let xki be the in-plane position
of the ith positive charge in the kth layer. In absence of external fields, the entropy of a
particular configuration of N defects is given by:
SN =
K
N
(
k=−K i=j
V (|xki − xkj |) −
N
V (|xki − xk+1
|))
j
(2.4)
i,j
Here V (R) = −κ ln(R/τ ), where τ is a hard-core radius defining the minimum separation
between two charges and κ = 1/2. The first term corresponds to the repulsion of positive
charges within the same layer. The absence of a factor of
1
2
in front of this term is due
to the string constraint: bringing two positive charges in the same plane close together
also involves bringing together their negative partners in the plane above. In terms of our
19
positive charge formulation, this means the repulsion is twice as large. The second term is
the interlayer interaction. Physically, a positive charge in layer k has a negative partner in
the layer k + 1 which attracts the positive charges in layer k + 1. In terms of our positive
charge formulation, like charges in the same plane repel one another but like charges in
neighboring planes attract.
We assume a variational mean field density of the form:
k
K
ρ(x−K
1 , ..., xi , ..., xN ) =
N
K ρk (xki )
k=−K i=1
(2.5)
N
which asserts that all particles in a given layer k have the same probability density ρk (x)/N ,
but the density may vary from layer to layer. We also need the normalizing condition:
d2 xρk (x) = N
(2.6)
A
This assures that each layer contains N particles: particles can not jump from layer to layer
in this formulation. This trial function implies a variational entropy functional:
Sρ,N
K 1
−
=
k=−K
−
d2 xd2 x (ρk (x) − ρk+1 (x))(ρk (x ) − ρk+1 (x ))V (|x − x |)
2
d2 xρk (x) ln(
ρk (x) )
N
(2.7)
where we have ignored terms of order 1/N . The first term is optimized if the density is
the same function in every layer: ρk (x) = ρk+1 (x). This leaves the second term, which
is maximized (given the normalization constraint) when the density is uniform ρk (x) =
N
A
which gives Sρ,N = (2K + 1)N ln A (A is given in units of τ ). To investigate the linear
response of the system, we may apply a perturbing potential to the objects in the k = 0
plane. In particular, we consider the effect of placing a positive charge at the origin of the
plane on this uniform density solution. The details of the calculation are given in Appendix
B but we may quote the result:
1
x
δρ( , k) = 2 2
ξ
4π ξ
−1/2 is·( x )
ξ
d2 s s2 (s2 + 2)
e
1 + s2 + 2 s2 (s2 + 2) 1 + s2 +
s2 (s2 + 2)
k−1
(2.8)
20
where the in-plane length scale is given by ξ =
A
4πκN
1/2
. We note first that this expression
diverges at small x for k = 0, which is not surprising because the assumption of a linear
response would be not be valid so close to the perturbing charge. The expression would be
valid at larger k and an interesting feature is that when x = ξ , the decay in the z-direction
does not depend on any physical parameters, i.e. there is no length scale in the z direction.
We will return to this point in the next section.
The correlation length has a temperature and field dependence contained in the factor
(N/A). We may estimate the number of defects, given a temperature and field, from the
partition function:
Z = e−βA =
y (2K+1)N
N
(N !)2K+1
eSN
(2.9)
where SN is the entropy of having N defects and y = e−Es /kB T is the fugacity of a positive
defect (y 2K+1 is the fugacity of a “string”) where Es = 8gµB JB/3 is the energy per kagometriangle bilayer of a string defect. At mean field level, we may replace SN by Sρ,N =
(2K + 1)N ln A. In Appendix B, it is shown that N ∼ yA so that we obtain the final
result:
2
ξ,M
F ∼ exp(8gµB JB/3kB T )
2.4.2
(2.10)
RG calculation
When the gas of defects is fairly dilute, we may expect that the screening is not effective
enough to justify a mean field treatment. In this section, we account for fluctuations
by making a real space renormalization group calculation using methods similar to the
Kosterlitz treatment of the 2d Coulomb gas[31, 33].
The dynamical objects described by Hamiltonian 2.4 are “dipoles of length 1”, i.e.
objects consisting of dipoles where the negative charge is rigidly connected to a positive
partner one (kagome) plane below. We need to generalize this model in order to do an
RG calculation. The generalization that we consider is allowing for dipoles of arbitrary
length. An “l-dipole” is an object where the negative charge lies directly l planes above
its positive partner to which it is rigidly connected. While the original problem involved
21
just the coupling of nearest neighbor planes, our generalized model involves all possible
couplings. Associated with each l-dipole is a fugacity yl /2π (the 2π is for convenience).
Fig. 2.7 is a cartoon of our generalized model. The grand partition function for the system
may be written as:
Z=
(yl /2π)N,l
{Nk,l } k,l
(Nk,l )!
Z[{Nk,l }]
(2.11)
where Nk,l denotes the number of l-dipoles in layer k; N,l is the number of l-dipoles in the
system; and Nk is the number of dipoles (of any length) that have their positive ends in
layer k. The sum is over all particle number configurations {Nk,l } that satisfy the charge
neutrality constraint in each plane: Nk =
l
Nk−l,l . The canonical partition function
corresponding to a given dipole distribution {Nk,l } is:
Z[{Nk,l }] =
(1)
(2)
(1)
(2) d2 xk,i d2 xk,i xk,i − xk,i Ωτ k,i
τ2
τ2
δ
τ
exp −H({Nk,l })
(2.12)
H({Nk,l }) is the Hamiltonian (actually an entropy) corresponding to the dipole distribution
(1)
{Nk,l }. The coordinate xk,i is the planar coordinate of the ith positive charge of layer k
(2)
and xk,i is the planar coordinate of its negative partner which lives in layer k + l(i), l(i)
being the length of the dipole being described. The string constraint is imposed by the
delta function, where we use the normalization
d2 x x
R2 τ 2 δ( τ )
= 1. The product is over all
positive charges in all layers. The integration is over the space Ωτ . This is defined to be
the set of all possible spatial configurations of the dipole distribution {Nk,l } that respect
the hard-core constraint: no two charges in a given plane may be closer than distance τ .
Our procedure is an extension of the treatment in Refs. [31],[33]. The first part of an
RG procedure normally involves integrating over the high momentum modes of the system.
In our problem, these correspond to configurations where in some plane, we have a pair
of charges separated by a distance between τ and τ + dτ . We assume a dilute system so
only oppositely charged pairs are considered and also the distance between the members
of a pair is taken to be much smaller than the distance from the pair to another charge.
These approximations were also used in [33] and are based on the Boltzmann factor for
configurations containing same charge pairs being much smaller than that for oppositely
22
charged pairs. Strictly, we would need to keep all of these other states unless they are
explicitly shown to be irrelevant in the RG sense but this demonstration is for another day.
The basic coarse-graining step in our RG transformation is illustrated in the lower part
of Fig. 2.7. Suppose a particular state involves pairing the negative end of an l1 -dipole in
layer k with the positive end of an l2 -dipole in layer k + l1 . Viewed at long length scales, we
effectively have an (l1 + l2 )-dipole in layer k. We will find that integrating over all possible
pairings gives a leading term (which just involves replacing Ωτ with Ωτ +dτ ) and a number
of correction terms of order dτ where two short dipoles were destroyed and replaced by a
longer dipole. Since the procedure respects the charge neutrality constraint, these correction
terms will combine with other terms in the grand partition sum. The second step involves
rescaling lengths so that the high momentum cutoff, in the new variable, is the same as
before. The aim is to see how the fugacities and couplings change as we run this procedure.
Details of the calculation are given in Appendix C. Here we give the resulting flow
equations:
dy1
dt
dyl
dt
dκ
dt
= (2 − κ)y1
(2.13)
= (2 − κ)yl +
l−1
ym yl−m
(2.14)
m=1
= 0
(2.15)
where t = ln τ . One notable feature is that the coupling does not change with the flow, even
at second order in the fugacity, in contrast with the 2d Coulomb gas[33] where it does vary.
This indicates that our defects are stiffer objects than usual charges. Another observation is
that for the initial conditions of our physical problem, namely that y1 (0) = y0 = 2πe−Es /kb T
and yl (0) = 0 for l > 1, the flow equations have an exact solution:
yl = y0 τ 2−κ
y 0
2−κ
l−1
(τ 2−κ − 1)
(2.16)
Our RG is valid as long as the corrections to the fugacities are small, meaning that the
derivatives dyl /dt should be bounded. If we look at the above result, Eq. 2.16, we see that
when the term in brackets is greater than 1, yl diverges with l. Therefore, a critical length,
23
which we interpret as an in-plane correlation length, is defined by when the term in brackets
equals 1:
y0
(ξ 2−κ − 1) = 1
2 − κ ,RG
(2.17)
Substituting earlier expressions and noting that for our system, κ = 1/2, we find that:
2
=
ln ξ,RG
32gµB JB ln(e−Es /kB T + 2 − κ) 1+
9kB T
Es /kB T
2
ξ,RG
∼ exp(32gµB JB/9kB T )
(2.18)
for the fields and temperatures of interest. This value is the same as that predicted in
Ref. [22] using a free energy argument. For τ < ξ,RG , yl decreases with l which means
that states with long dipoles are less probable than states with short dipoles. If τ > ξ,RG ,
yl diverges with l which suggests that longer dipoles are favored, but, as mentioned above,
the RG procedure is no longer valid in this regime. We note that when τ = ξ,RG , yl
is independent of l so that, as in the mean field calculation discussed above, there is no
discernible length scale in the z direction.
If τ < ξ,RG , then we may consider an out-of-plane length scale, which we define nominally as the value of l = lτ for which yl /y1 = 1/e.
1
lτ = 1 +
ln
ξ 3/2 −1 ,RG
(2.19)
τ 3/2 −1
We may interpret lτ as the typical length of a string segment that is captured by a tube of
diameter τ (where a tube need not be straight).
2.4.3
Comparison with simulation
In Fig. 2.8, we show the magnetization as a function of the magnetic field strength on a
log-log scale, computed by Isakov[34] using an efficient cluster algorithm. The magnetization
should scale with the average density of defects, which in turn should scale like the inverse
square of the in-plane correlation length. As shown in this figure, the data at low fields
are well fit by the exponent 8/3 obtained in the mean field calculation discussed earlier.
At somewhat higher fields, the data are well fit by the exponent 32/9, obtained by the RG
24
calculation discussed earlier and also in Ref. [22] by looking at the entropic contribution
to the free energy. At high fields, the exponent of 8L/3 (=16 for L=6 (sites), as was the
case in the simulations) characterizes a regime where finite-size effects are important, as
discussed below.
The low field crossover makes qualitative sense in that at low fields, there will be many
defects which screen one another which suggests that a mean field treatment may be reasonably accurate. At higher fields, the gas of defects is more dilute so an RG treatment
would be required.
The high field crossover is a finite-size effect since the position of a crossover between
exponents is system size dependent and the corresponding exponent is also system size
dependent, getting steeper with increasing system size. The finite-size behavior, alluded
to in Section 2.3, may be explained as follows. At high magnetic fields, there are a small
number of string defects in the system. The magnetization and the energy of one string
defect in a system of size L are −4LgµB J/3 and 4LgµB JB/3 respectively. The energy cost
grows linearly with system size and, as mentioned above, the defects are favored solely due
to their entropic contribution to the free energy. At sufficiently high magnetic fields, a given
system will be too small to provide the entropy to balance the energy cost of a string. This
will occur when the magnetization per spin reaches the magnetization of a system with one
string defect:
m =
=
1/3 − 2(4L/3)/(16L3 ) gµB J
1/3 − 1/(6L2 ) gµB J.
(2.20)
In this case, the statistical weight of a single string defect will be a Boltzmann factor
exp(−8LgµB JB/3kB T ) and the magnetization will equal:
m = [1/3 − C exp(−8LgµB JB/3kB T )]gµB J
(2.21)
where C is some constant. The crossover between different regimes occurs when the magnetization reaches (2.20). We have good agreement with the 8L/3 behavior for a variety of
system sizes, including L = 6 which is shown in Fig. 2.8.
25
2.5
The high field regime
The physics near the transition may be approximately described by the following Hamiltonian which acts on the kagome lattice:
H
=
K(si , sj )si sj − h
si ,
T
i
ij
(2.22)
where the sum is over all nearest neighbors; si are classical Ising spins taking values +1 and
−1; h is the strength of a fictitious magnetic field; and K(1, 1) = 0, K(1, −1) = K(−1, 1) =
K = [gµB JB/6 − Jeff ] /T , and K(−1, −1) = ∞. The coupling constants imply that each
triangle of the kagome lattice contains at most one down pseudospin and that down spins
cost energy (positive or negative dependent on the magnetic field strength). In terms of
the dimer representation, this approximation involves considering only states where each
site contains one dimer or zero dimers (i. e. a monomer). States involving sites with two or
three dimers involve an infinite energy cost so are ignored.
Isakov[34] calculated the magnetization and entropy within this simplified monomerdimer model using the Bethe approximation[28]:
1 1
2 1 + x2
1
2z 3
3xz ln z
+ ln 2
,
S = −
2 + 6xz 4 x (3z − x)
m =
where x = 2z/(1 +
√
(2.23)
(2.24)
1 + 8z 2 ) and z = exp(−2K).
In Fig. 2.9, we compare these expressions with a Monte Carlo simulation of Hamiltonian
2.2. In this simulation, sites containing two or three dimers are given the appropriate
Boltzmann weight instead of being explicitly ignored. The simulation is of a kagome lattice
with 16x16 up-triangles (768 total spins). The standard single spin-flip Metropolis algorithm
was used, which may explain the inaccuracy in the simulated entropy at low fields, where
a more clever scheme may be needed to sample the degenerate manifold. The entropy was
computed, for a given field, by integrating from high temperatures (where S/kB = (3/4) ln 2
per atom) to low temperatures. The agreement of the simulation with Isakov’s analytic
expression validates the monomer-dimer picture of the transition.
26
There is a giant peak in the entropy at the transition point, S/kB = 1/4 ln(16/5) ≈
0.291, which exceeds even the zero field entropy. The peak is due to the crossing of an
extensive number of energy levels which have macroscopic entropies. For B = Bc , the
energies of states corresponding to different numbers of monomer defects are equal since
the monomer and dimer weights are, by definition, equal at the critical field. There are an
extensive number of states corresponding to a given number of monomers (below saturation). So while the [111] magnetic field selects a subset of the zero field spin ice ground
state manifold, near the critical field, the ground state space also includes a large number
of states not in the ice manifold. Our model predicts that these extra states more than
compensate causing a peak which exceeds even the zero field entropy.
2.6
Crossing points
The theory described in the previous section implies that the curves of magnetisation versus
field, plotted for different temperatures, will display a crossing point. This arises simply
because the partition function depends on magnetic field and temperature effectively only
through the combination (B − Bc )/T . Thus, when plotted as a function of B − Bc , the
curves coincide only at the point B = Bc . At this point, the Bethe approximation gives a
value for the magnetisation of m = 0.4gµB J.
In addition, we expect a crossing point at low fields, due the interplay of string and
monomer defects. Indeed, where the plateau is well-formed, the string density is ns ∼
exp(−32gµB JB/9kb T ) and the monomer density is nm ∼ exp(−8Em /7kB T ), where E =
gµB J(Bc − B)/3 is the energy of creating one monomer. The crossing point occurs when
ns = nb . With logarithmic accuracy, we can write
8gµB J(B − Bc )
32gµB JB
=
.
9kb T
21kB T
Thus the crossing point lies at B = 3Bc /31.
(2.25)
27
2.7
Relation to experiment, other theories, and applications
Our model gives a description of the high field transition that is qualitatively consistent
with experiment for a range of temperatures [11]. The predicted value for the entropy of the
lower [111] plateau, based on the exact calculation for classical dimers on the honeycomb
lattice[21, 23], has been confirmed in experiment[11]. The magnetization curves cross at
a field close to the predicted critical field[11] and in particular, a peak in the entropy has
been recently observed[15]. The height of this experimental peak is smaller than the zero
field entropy, while our simple model predicts a larger peak.
However, a more serious discrepancy is a recent experiment[35] on the spin ice compound
Dy2 Ti2 O7 which strongly suggests that at low temperatures, the high field transition becomes first order. In Ref. [35], the onset of first order behavior, indicated by discontinuous
magnetization curves and hysteresis, was found to occur for temperatures lower than a
critical temperature of Tc ∼ 0.36K (∼ 0.327Jef f,Dy /kB ). Fig. 2.9 shows that our predicted
curves remain continuous even at temperatures below this observed Tc .
Our initial hypothesis was that this discrepancy is due the long range nature of the
dipolar interaction, which we approximated as a nearest neighbor Ising model. In Section
2.2, we discussed recent work[17, 18, 19, 20] that demonstrated that this nearest-neighbor
model has the same ground state manifold as a model dipolar interaction. However, this
does not imply a correspondence between the excited states of these models. In the nearestneighbor case, the natural excitations at high magnetic fields are monomer defects and the
proliferation of these defects appears to describe the regime where the magnetization curves
are continuous. In the regime where the magnetization is discontinuous, the excited states
of the full dipolar interaction may differ significantly from the excited states of the nearestneighbor model.
The simplest way to account for the long-range dipolar interaction is to model the
further neighbor interaction terms as giving rise to a magnetic field proportional to the
magnetization. By assuming the magnetization M , as a function of the effective field
B + αM , has the same functional form as given in figure 2.9, we may self-consistently
28
determine M for a given B. Using α as a free parameter, we find that this simple model
predicts the onset of first order behavior, at the experimentally observed critical field Bc ,
only for temperatures in the millikelvin range. To obtain a higher numerical Tc requires a
larger α, which causes a lower numerical Bc . To get the numerical Tc to match experiment
requires an α so large that our numerical Bc is “negative” (in the sense of artificially
extending the M = 1/3 line of figure 2.9 for the purpose of a spline fit). Therefore, it appears
that the role, if any, of long-range interactions in explaining this feature is something more
subtle.
Another possibility which we have not considered is the impact of the slowdown of the
dynamics which is observed at low temperature.[36] The hysteresis in the magnetization[35]
at low temperatures, which was interpreted as indicating a first order transition, may be a
nonequilibrium effect.
As for the crossing points mentioned above, the high-field one does indeed appear to
be present in the experimental data[10, 35] in the appropriate temperature range. The
experimental value of the magnetization at the crossing point is about m = 0.38gµB J,
reasonably close to the theoretical value m = 0.4gµB J. By contrast, a crossing point
at small fields is harder to make out, and an approximate estimate of its location gives
B = 0.35Bc , in disagreement with the theoretical B = 3Bc /31.
2.7.1
Cooling by adiabatic (de)magnetization
At low temperatures, near the degeneracy point, the partition function depends on magnetic
field and temperature effectively only through the combination (B − Bc )/T . One may thus
argue that the spike may be used to effect cooling by adiabatic demagnetization[37] in
exactly the same way one may use paramagnets – analogous constraints limit the application
in either case.
There are two features which may be worth pointing out at this point. Both follow from
the fact that – unlike in the case of a paramagnet – Bc = 0. Firstly, maximal cooling occurs
at a finite field, namely around Bc . This phenomenon may therefore be useful to effect
29
cooling for a magnet in a field, with the restriction that Bc , for a given spin ice compound,
is not tunable. Secondly, if B approaches Bc from below, one can in fact obtain “cooling
by adiabatic magnetization”, as entropy and magnetization grow together in this regime.
2.8
Conclusions
In this paper, we have analyzed in detail the magnetization curve of nearest-neighbor spin
ice in a [111] magnetic field. The basic ingredient which makes this system particularly
interesting is that a uniform field can be used to couple to the Ising pseudospins as a
staggered field[38, 39]. This amounts to the possibility of applying fields which would have
appeared to be rather unnatural in the formulation of a simple Ising model (without the
detour via spin ice) on the pyrochlore lattice.
As a result, one observes an attractively rich behavior. Perhaps the most salient is the
dimensional reduction from pyrochlore to kagome under the application of an external field.
The restoration of three-dimensionality upon weakening the field goes along with the string
defects. We hope that the extension developed here of Kosterlitz’s RG treatment to such
extended defects might be of more general use.
A particularly attractive feature of the monomer-dimer model we have obtained here
lies in the fact that the relative monomer and dimer fugacities in the low-temperature
(T Jeff ) regime are given by simple Boltzman weights of Zeeman energies. They are
thus straightforwardly tunable by changing the strength of the applied field. In particular,
anisotropic fugacities for the dimers can be obtained by tilting the field, and they therefore
do not require an actual manipulation (such as an application of anisotropic stress) of the
two-dimensional layer.
As discussed previously in Ref. [22] the price for our ability to analyze the model in such
detail has been the omission of the long-range nature of the dipolar interaction. A truncation
of the interaction at only the nearest-neighbor distance would seem a rather drastic step; an
expectation of quantitative agreement between experiment and the nearest-neighbor model
will in general likely be misplaced. Recent work has shown that this approximation is not
30
entirely unreasonable[19, 20] and this observation might lie at the basis of the fact that
the measured dipolar ice entropy agrees so well with Pauling’s estimate. Our ‘prediction’
of the entropy peak between the intermediate and saturated plateaux bears witness to the
promise of our approach to unearth at least some qualitative features of interest.
Magnetization fraction
31
monomers
1/2
dimers
1/3
dimer-monomer
strings
1/6
linear response
0.25
0.50
0.75
1.0
1.5
1.0
1.5
Magnetic field (T)
S/k (per spin)
0.3
0.2
0.1
0.25
0.50
0.75
Magnetic field (T)
Figure 2.3: Properties of spin-ice as the [111] magnetic field is varied. These curves are for
illustration and do not show actual numerical or experimental data. We have indicated the
regions where various analytic approaches apply. The linear response regime is not directly
discussed in this thesis; for a detailed treatment of that regime, the reader should consult
Refs. [1, 34].
32
H [111]
Figure 2.4: The figure depicts the pyrochlore lattice in profile with the [111] direction
indicated. The lattice sites are indicated by the black dots and the structure of interwoven
kagome and triangular planes is clearly seen. Each triangular spin is a member of an up
and down pointing tetrahedron and will point along or against the field.
33
H [111]
Figure 2.5: A single string defect. The figure depicts the pyrochlore lattice in profile with the
[111] direction indicated. The large black dots indicate that the corresponding triangular
spin is oriented against the field. This causes the 2-up, 1-down “kagome ice” rule to be
violated on the triangles immediately above and below the flipped spin. The resulting
defects are indicated by the circles. As discussed in the main text, a pair of defects is
created in each layer. A defect which is the base of an up (down) pointing tetrahedron
is called “positive” (“negative”). In each plane, the two defects may move independently
but the string constraint requires each positive defect to be rigidly connected to a negative
defect in the layer above. This condition is denoted by the thick bars.
34
+
+
+ -
++
+
+
+
+
- +
+ +
+
- +
+ +
++
+
+
+ -
-
+
- +
+
+ -
+ +
-
+
+ +
+
+
- +
+
Figure 2.6: Kagome ice. The constraint on the [111] plateau is that each triangle of the
kagome lattice must have two up spins and one down spin. The centers of these triangles
form a honeycomb lattice as indicated by the dotted lines above. Draw a dimer (indicated
by thick lines) on every link containing a down spin. The two up, one down constraint
ensures this procedure will place one and only one dimer on every site. Hence the kagome
ice manifold maps onto dimer coverings of the honeycomb lattice.
35
+
-
-
-
-
+
+
-
+
-
+
+
-
+
-
- +
+
+
Figure 2.7: Renormalization group analysis of string defects. In the physical model, the
dynamical objects are oppositely charged pairs, where the negative charge lies in the plane
directly above its positive partner, to which it is rigidly connected. Our generalized Hamiltonian considers objects of arbitrary length. The basic coarse-graining step in our RG
transformation is shown in the lower part of the figure: smaller objects grow into larger
objects at the expense of renormalizing the fugacities and coupling.
36
1/3-m
10
0
10
-2
10
-4
10-6
10
10
MC L=6
0.3539 exp(-8/3 x)
0.6278 exp(-32/9 x)
7
6.16 10 exp(-16 x)
-8
-10
0
0.5
1
1.5
2
2.5
3
0.4
0.5
0.6
1/3-m
gµBJB/kBT
10-1
MC L=6
0.3539 exp(-8/3 x)
0
0.1
0.2
0.3
gµBJB/kBT
1/3-m
10-1
10-2
MC L=6
0.6278 exp(-32/9 x)
10-3
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
gµBJB/kBT
Figure 2.8: The crossover between exponents. The magnetic field on the plateau is too
weak to create a string defect. As the field is lowered, strings nucleate and the 32/9 line
describes the dilute string limit quite accurately. The RG assumes a dilute gas of strings
and, expectedly, breaks down at lower fields where strings proliferate. In this regime, a
mean field treatment works fairly well as indicated by the 8/3 line.
37
0.52
0.48
m/µ
0.44
0.4
kT/Jeff = 0.1
kT/Jeff = 0.3
kT/Jeff = 0.5
Bethe approximation
0.36
0.32
-4
-2
0
2
4
6
8
10
gµBJ(B-Bc)/3kBT
0.3
kT/Jeff = 0.15
kT/Jeff = 0.3
kT/Jeff = 0.5
Bethe appr.
Series exp.
Exact result
Pauling result
0.25
S/kB
0.2
0.15
0.1
0.05
0
-10
-5
0
5
10
15
gµBJ(B-Bc)/3kBT
Figure 2.9: The magnetization (top) and the entropy (bottom) around the transition between the plateaux. The Bethe approximation and higher order series expansion calculations
of Isakov[34] are compared to the Monte Carlo results. The exact result for the entropy
at zero monomer density and Pauling’s estimate for the entropy at zero magnetic field are
shown for reference.
Chapter 3
SU(2) Invariant spin 1/2
Hamiltonians with RVB and other
valence bond phases.
3.1
Introduction
Just over thirty years ago Anderson[40] introduced the resonating valence bond (RVB) state
as an alternative to Néel ordering in antiferromagnets with strong quantum fluctuations.
In essence, he proposed that on a sufficiently frustrated lattice an S = 1/2 system would
exhibit a disordered state at T = 0, which would be captured by a wavefunction of the form
|ψ =
Ac |c
(3.1)
c
where |c is a configuration of singlet pairings of spins or valence bonds (Fig 3.1). For
sufficiently short-ranged valence bonds this describes, in contrast to the Néel state, a state
with short-ranged spin correlations.
The discovery of the cuprates and the suggestion that their superconductivity could be
traced to RVB physics[41] greatly energized the elucidation of the RVB idea and by now
a rather complete understanding of its internal logic has emerged. In modern parlance,
38
39
Figure 3.1: Sample valence bond configurations. The thick lines represent singlet pairings.
The original formulation allowed for bonds of arbitrary length (left). Considerable progress
has been made by restricting to configurations with only nearest-neighbor valence bonds
(right).
an RVB phase is a topological phase, characterized by excitations with fractional quantum
numbers and a low energy gauge structure which mediates topological interactions among
the excitations[42]. The excitations include spinons, the S = 1/2 excitations produced by
breaking a valence bond, as well as collective excitations within the valence bond manifold
(see figure 3.2). The sr-RVB[43] (short ranged RVB) with short-ranged bonds and gapped
spinons will be our concern in this paper. A post-cuprate version with longer-ranged bonds
and gapless spinons [44] has also been the subject of recent progress.[45] Readers familiar
with one dimensional lore will note that the short-ranged and long-ranged RVBs generalize
the physics, respectively, of the Majumdar-Ghosh chain[46] and the Bethe chain [47] to
higher dimensions.
The important progress that we have described has been kinematical. It has not directly answered the question of realizing RVB phases for actual Hamiltonians. The original
proposal was made for the nearest-neighbor Heisenberg model on the triangular lattice but
that is now generally believed to exhibit weak Néel order[48]. To make progress on the
dynamical front, Rokhsar and Kivelson[49] introduced the quantum dimer model (QDM)
which assumes that the low energy dynamics is dominated by valence bond configurations
40
x
Figure 3.2: Typical excitations of an RVB liquid. The left figure depicts a pair of spinons,
fractionalized excitations with S = 1/2, formed by breaking a valence bond. Spinons are
unpaired spins which move in the RVB liquid background; the interaction between spinons
depends on the lattice geometry. The right figure depicts a vison, which is an excitation
within the valence bond subspace. If we consider an RVB liquid which is an equal amplitude
superposition of nearest-neighbor valence bond states, then the vison is defined by the
wavefunction |ψv = c (−1)Nc |c where Nc is the number of bonds which cross the dashed
line shown in the figure. Clearly this state is orthogonal to the RVB state, |ψ = c |c.
The interaction between spinons and visons is discussed in detail in Ref. [42].
of short range which are taken to be nearest-neighbor in the versions studied to date. Such
configurations are labelled by dimer coverings of the lattice at issue and the quantum dimer
Hamiltonian acts in a Hilbert space spanned by such coverings. The program of studying
the simplest dimer models has been rather fruitful. It is now clear that Z2 RVB phases
may arise on non-bipartite lattices in d ≥ 2[51, 52] while bipartite lattices in d > 2 give
rise to U (1) RVB phases that exhibit a gapless “photon”[53, 54]. In addition a variety
of crystalline phases have been identified, most notably a Cantor deconfined region[55] of
interleaved commensurate and incommensurate valence bond crystals on bipartite lattices
in d = 2.
The next order of business then, is to find rotationally invariant, local, spin Hamiltonians
that are accurately described by these well understood dimer models. This is the problem
that we solve in this paper thus completing a program initiated by Chayes, Chayes and
Kivelson[56]. The strategy we follow is that of constructing Klein Hamiltonians[57] with
41
large energy scales which select nearest neighbor valence bond states as their ground states,
separated by a gap from excited states. We then lift this degeneracy by the inclusion of
perturbations that precisely mimic the terms in the quantum dimer models of interest.
To control this procedure, with its difficulties stemming from the non-orthogonality of the
valence bond basis, we introduce a parameter which is the extent of decoration of the
reference lattice. By suitably tuning this parameter we are able to make our dimer model
realizations arbitrarily accurate. An elegant feature of this limit is that it enables us to
establish the existence of a spin gap about the nearest-neighbor valence bond manifold and
to discuss the states above this gap in terms of “microscopic” spinons whose meaning will
become clear below.
We note that our tuning procedure occurs within the space of SU(2) invariant Hamiltonians. This is in contrast to approaches involving enlarging the symmetry group to Sp(N )
or SU(N ) and studying the large N limit[58]; in these cases, the applicability of results to
Sp(1) ≡ SU(2) is not obvious.
This is also a good place to note that there is a considerable body of work on variational[59]
and finite-size studies of some of the two-dimensional phases discussed in this paper, e.g.
the early finite-size study of a multiple-spin Hamiltonian on the triangular lattice that adduced evidence for a topologically ordered phase[60]. A comprehensive review of such work
is given in Ref. [61]. This work is complementary to ours as it deals with somewhat simpler
Hamiltonians but is unable to access the thermodynamic limit in a controlled fashion. We
also note that, there is a large and growing literature on more general models with topological phases which we skip in our focus on S = 1/2 spin systems; this work also finds
inspiration from the proposal that a quantum computer may be robustly created from a
topological phase[62]. Finally, we note that there are encouraging reports of spin liquids in
experimental systems[63, 64].
In the rest of this chapter we give details of our constructions. We begin with a quick
review of quantum dimer models and the known results on their phase diagrams in Section
3.2. In Section 3.3 we explain our strategy with the honeycomb lattice serving as an example;
42
this realizes the physics of bipartite dimer models in d = 2. In Section 3.4 we show how
the physics of non-bipartite dimer models in d = 2 and bipartite and non-bipartite dimer
models in d > 2 may be obtained from spin models. In Section 3.5 we discuss two spin
models on the non-bipartite pyrochlore lattice, one exhibiting a Z2 RVB phase and the other
a U (1) RVB phase. We conclude with a summary (Section 3.6) and a set of appendices
that contain some technical material.
3.2
Quantum dimer models
For a system of spins si on a lattice Λ, a (nearest-neighbor) valence bond state is a product
wavefunction of the form Ψ =
ij ψij
where ψij =
√1 (ψ ↑ ψ ↓
i j
2
− ψi↓ ψj↑ ) and the product is
over nearest-neighbor pairs (i, j). The product is defined so each spin forms a singlet with
exactly one of its neighbors. Each valence bond state corresponds to a hard-core dimer
covering of Λ where a dimer connecting two sites corresponds to a singlet bond between
the respective spins. Valence bond states are not orthogonal but the overlap between two
arbitrary states is exponentially small in the length of closed loops obtained by superposing
them. This suggests that they are linearly independent on sufficiently open lattices and
indeed there are proofs for the square and honeycomb lattices[56] and numerical evidence
that this is so even on the triangular lattice[65].
The identification with dimer coverings suggests that any low energy dynamics restricted
to the valence bond manifold can be represented by a quantum dimer Hamiltonian acting
on orthogonal dimer states. More precisely, the set of hard core dimer coverings of the
underlying lattice may be taken as an orthonormal basis spanning a dimer Hilbert space;
the orthonormality of the dimer coverings defines the inner product on this space. We may
introduce dynamics on this dimer Hilbert space and the simplest such Hamiltonian on the
square lattice, written down by Rokhsar and Kivelson [49], has the pictorial form
H
= ∑-t( 〉〈
QDM
+h.c.)+v( 〉〈 + 〉〈
)
(3.2)
where t and v are positive constants and the sum is over all possible square plaquettes.
43
Evidently, this can be supplemented by kinetic energy terms which act on longer loops and
potential energy terms which count more complicated dimer motifs.[66] Fig. 3.3 outlines
the appropriate generalizations of these basic operators to the honeycomb and triangular
lattices.
The passage from valence bonds to dimers, however, has to contend with two complications. One is that one needs to choose a phase convention for the valence bonds, which
is subject to restrictions on what signs one can obtain for various couplings in the dimer
Hilbert space. The other, already alluded to, is the lack of orthogonality of the valence
bond states which makes the transcription from a spin model to a dimer model (and the
relationship between the respective Hilbert spaces) non-trivial. We will deal with both
problems later in the paper; here we merely wish to alert the reader to their existence.
An important property of the quantum dimer Hamiltonian is the existence of the
“Rokhsar-Kivelson point” (RK point) t = v, where any equal amplitude superposition of
all dimer coverings connected by the operation of the kinetic energy is a ground state. To
see this, note that for every flippable plaquette, the second term gives a penalty v while the
first term gives at most a benefit of −t. Nonflippable plaquettes have zero energy according
to HQDM . This gives a lower bound for the ground state energy: E0 ≥ min{0, NP (v − t)},
where NP is the number of plaquettes in the lattice. The equal amplitude state has energy
nf l (v − t), where nf l is the average number of flippable plaquettes in the state. At
v = t, this saturates the lower bound and, since the equal amplitude state is an eigenstate
of HQDM (at v = t), we may conclude that it is a ground state when v = t.
Thus the ground state correlations at the RK point reduce to those of solvable classical
dimer models. Additionally, the infinite temperature static correlations of QDMs also reduce
to those of the same classical models. These features, along with the additional one that
Hamiltonians of the form (3.2) can be simulated by Monte-Carlo without any sign problems,
have been crucial to making progress in determining the phase diagrams of the quantum
models.
As a consequence of this progress we now know that:
44
i) QDMs on bipartite lattices in d = 2 do not exhibit an RVB phase. The equal amplitude
state present at the RK point, v/t = 1, does not extend into a phase. As v/t increases,
the system generically passes through a sequence of interleaved commensurate and incommensurate crystalline phases before reaching the staggered valence bond solid (VBS) phase.
These incommensurate ground states, whose measure approaches unity near the RK point,
turn out to have deconfined monomers, a phenomenon coined Cantor deconfinement[55].
Some more details about Cantor deconfinement are given in Appendix A. As v/t decreases
from unity, the system passes through a plaquette phase[67] before undergoing a first-order
transition to a columnar VBS. The phase diagram for these systems is given in Fig. 3.4.
ii) QDMs on non-bipartite lattices in d = 2 may exhibit RVB phases. These are Z2 RVB
phases captured by a purely topological BF theory[42]. For the triangular lattice, it has
been shown[51] that for v/t > 1, the system is in a staggered VBS; for v/t ≤ 1, there is a
deconfined RVB liquid phase. As v/t is further reduced, there are probably a small number
of VBS phases culminating in the columnar state. The phase diagram for these systems is
given in Fig. 3.5.
iii) QDMs on non-bipartite lattices in d = 3 and higher also exhibit a Z2 RVB phase[52].
iv) QDMs on bipartite lattices in d = 3 and higher exhibit a U (1) RVB phase with a gapless,
linearly dispersing transverse mode, the “photon”[53].
We now turn to the task of constructing spin models whose low energy dynamics is
precisely captured by these dimer models. We begin, for pedagogical specificity, with the
honeycomb lattice.
3.3
Honeycomb lattice: Bipartite physics in d = 2
Our strategy for realizing dimer models proceeds in three steps. First, we construct, following Klein, a local spin Hamiltonian that has valence bond states as its ground states.
Next we perturb it to obtain a QDM. Finally, we decorate the lattice to simplify the QDM
to the well studied form (3.2). In the high decoration limit, we show the existence of a gap
and give a description of the spectrum in terms of spinons.
45
3.3.1
Klein model
The basic idea of the Klein model consists of considering a cluster of z sites (typically, a
spin and its z − 1 neighbors) and deterring, via an energy penalty, this cluster from having
maximal total spin Stot = z/2. If two of the spins in the cluster form a singlet bond, this
condition is satisfied. This is why Klein Hamiltonians naturally lead to valence bond ground
states.
In particular, for a system of spins si on a lattice Λ, a Klein Hamiltonian is a sum of
projection operators P̂N (i) defined as follows. For each site i, consider the neighborhood
of spins N (i) consisting of the spin at site i and its (zi − 1) nearest neighbors. Let P̂N (i)
be the projector onto the highest total spin state of the cluster. The Klein Hamiltonian is
formally given by the expression:
HK =
P̂N (i) ,
(3.3)
i∈Λ
with total spin of cluster N (i) given by:
N (i) =
S
sj
(3.4)
j∈N (i)
We may write P̂N (i) in terms of this operator. For example, if zi is even, then:
zi /2−1
P̂N (i) = Ci
2
SN
(i) − L(L + 1) .
(3.5)
L=0
The total spin of this cluster will take values from 0, 1,. . . ,(zi /2) − 1,zi /2. The factors of
this product are operators which sequentially annihilate all but the highest spin sector. The
form of the operator implies that larger clusters involve higher-order spin interactions– but
they always remain local.
If the constants Ci in Eq. 3.5 are chosen to be positive, then HK will have non-negative
eigenvalues. By construction, valence bond coverings are zero-energy ground states of HK .
This is readily seen by noting that the highest spin sector of a cluster is necessarily symmetric
under interchange of spins; if two of the spins in a cluster are paired in a singlet, then the
state is antisymmetic under their interchange so hence has zero projection in the highest
46
spin sector. We note that HK may also have zero-energy ground states outside of the
valence bond manifold. For the honeycomb lattice, there are proofs[56], for a restricted
set of boundary conditions, that the ground state manifold of HK is spanned by the set of
valence bond coverings. For other lattices, we can explicitly construct many non-valence
bond ground states (for example, Fig. 3.11). We will return to this question later but for
now, we assume that valence bond states span the ground state manifold of our lattice.
For the honeycomb lattice, Chayes et al.[56] have already written down the explicit form
of the Klein Hamiltonian in terms of spin operators. Their expression, up to unimportant
overall constants, is:
H =
si · sj +
i,j
+
1 si · sj
2 i,j
2
(
si · sj )(
sk · sr )
5 ijkr
(3.6)
The first and second terms are over nearest and next nearest neighbors respectively.
The third term is over quartets ijkr where i and j are nearest neighbors; k is a neighbor
of i different from j; and r is a neighbor of j different from i. A striking feature of this
Hamiltonian is that the leading term is the usual Heisenberg antiferromagnet.
3.3.2
Perturbations
We will now perturb the Klein Hamiltonian to obtain a QDM with dynamics. In doing so
we will use the overlap expansion invented by Rokhsar and Kivelson[49], which is predicated
on the linear independence of the valence bond states. That valence bond states on the
honeycomb lattice are linearly independent was proved in Ref. [56].
For the purpose of obtaining the dimer kinetic energy, it is sufficient to consider including
just an additional nearest neighbor interaction[68],
δH =
si · sj
(3.7)
i,j
To first order in degenerate perturbation theory, we may write this as an effective operator
on the valence bond states. First, we define an orthonormal basis set {|α} in terms of the
47
linearly independent valence bond states {|i}:
|α =
(S −1/2 )α,i |i
(3.8)
i
Here Sij = i|j is the overlap matrix element between valence bond states |i and |j.
The magnitude of the overlap of two valence bond states may be determined by overlaying
the two configurations forming what is called the transition graph[49]. The construction is
described in Fig. 3.6. As shown in Fig. 3.6, the transition graph consists of double bonds,
where the two states have a bond in common, and closed loops of varying (even) lengths.
The magnitude of the overlap Sij is given by 2Nl
i
xLi where Nl is the number of loops;
the product is over all such loops; Li is the length of loop i; and x =
√1 .
2
The sign of Sij
depends on how we choose to orient the bonds. By orientation, we refer to the fact that a
bond between sites 1 and 2 may be interpreted as the singlet bond ψ12 =
√1 (ψ ↑ ψ ↓ − ψ ↓ ψ ↑ )
1 2
1 2
2
or as ψ21 = −ψ12 . The key idea of the overlap expansion is to treat x as a small expansion
parameter.
We may specify the matrix elements of our effective operator in terms of the {|α} basis:
Hαβ = (S −1/2 δHS −1/2 )αβ
=
(S −1/2 )αi i|δH|j(S −1/2 )jβ
(3.9)
(3.10)
ij
If either state |i or |j contains the bond (12), then i|
s1 · s2 |j = − 34 i|j. If neither |i nor
|j contains the bond (12), then we have a non-zero matrix element only if spins 1 and 2 are
members of the same loop in the transition graph. If that is the case, then one may show
that i|
s1 · s2 |j = (−1)n/2 (∓ 34 )i|j where n is the length of the loop for the case where
spins 1 and 2 are separated by an even (odd) number of sites.
We now specialize to the honeycomb lattice. As sketched in the Appendix, we may orient
the bonds on the honeycomb lattice so that for any two states differing by a (minimal) length
6 loop, the overlap i|j has a positive sign. On the honeycomb lattice, the sign is a matter of
convention and we could have chosen the negative sign. Given the positive sign convention,
we conclude that our matrix element is given by:
i|δH|j = −
3
3nd
δij − (2x6 )ij + O(x10 )
4
4
(3.11)
48
where nd is the number of bonds (half the number of sites) and ij is a matrix that is 1
if states |i and |j differ by a length 6 loop and zero otherwise. We may also expand the
overlap matrix:
Sij
(S −1/2 )ij
= δij + 2x6 ij + O(x10 )
(3.12)
= δij − x6 ij + O(x8 )
(3.13)
Comparing the previous line with Eq. 3.8, we see that within the overlap expansion, each
|α has a largest component corresponding to a unique valence bond state. Therefore, we
will refer to the orthogonal set {|α} as the set of dimer states corresponding to the valence
bond coverings. In writing our effective operator, we absorb the leading term involving the
number of dimers nd times the unit operator into our definition of zero energy.
Hαβ =
[δαi − x6 αi + O(x8 )]
ij
3
× [− (2x6 )ij + O(x10 )]
4
× [δjβ − x6 jβ + O(x8 )]
(3.14)
3
≈ − (2x6 )αβ + O(x10 )
4
(3.15)
We conclude that the leading term in the overlap expansion is an operator with nonzero
matrix elements only between dimer states differing by a minimal length 6 loop. All of
these nonzero elements have the same value −t = −(3/4)(2x6 ). Thus we have obtained the
kinetic energy operator in the quantum dimer model (QDM). Note that we can conclude
this only because we were able to define a bond orientation convention such that all minimal
overlaps come with the same sign. Otherwise some off-diagonal terms would have energy
t and the energy bound arguments which we gave previously to conclude that there is an
RVB state at the RK point would no longer hold.
To obtain the potential energy term as the leading order effect, we need a more complicated interaction which we take to be:
δH = J
ij
si · sj +
49
v
(
s1 · s2 )(
s3 · s4 )(
s5 · s6 )
+(
s2 · s3 )(
s4 · s5 )(
s6 · s1 )
(3.16)
where the first sum is over nearest neighbors and the second sum is over elementary plaquettes (see Fig. 3.7).
The first term of Eq. 3.16 gives the QDM kinetic energy. Similarly, consider the operator
s12 s34 s56 = (
s1 · s2 )(
s3 · s4 )(
s5 · s6 ). If valence bond states |i and |j contain the bonds
(12), (34), and (56), then i|s12 s34 s56 |j = (− 34 )3 i|j. If state |i contains all three bonds,
then the diagonal matrix element i|s12 s34 s56 |i is (− 34 )3 . If state |i is missing one or more
bonds, then the diagonal matrix element is zero unless |i contains the three complementary
bonds (23), (45), and (61). In this case, the expectation of the operator equals (− 34 )3 x6 ,
which is higher order in the overlap expansion. It may be shown that off-diagonal matrix
elements evaluate to a term proportional to the overlap of the states, the proportionality
constant being of order unity. These results imply that the matrix element of the ring
interaction between two valence bond states is given by:
3 3
−
4
vnf l,i δij + O(vx6 )
(3.17)
where nf l,i is the number of flippable hexagonal plaquettes in configuration |i. We may
write this in terms of dimer states, as discussed above. Absorbing numerical factors into
the constants J and v, we arrive at our effective dimer Hamiltonian:
Hαβ = −Jx6 αβ + vnf l,α δαβ + O(vx6 + Jx10 )
= −tαβ + vnf l,α δαβ + O(vx6 + tx4 )
(3.18)
where t = Jx6 and nf l,α is the number of flippable plaquettes in the valence bond state
corresponding to dimer state α. If t and v are of order unity, then the higher order terms will
be small compared to the first two terms, which act on our dimer states (which are really
spin states) in a manner analogous to the QDM kinetic and potential energy operators on
usual dimer states.
50
√
For the actual problem at hand, x = (1/ 2) is less than 1 but is by no means tiny.
Hence the neglect of other terms induced by our perturbations is not obviously justified.
While we do not need them to be zero, we do need them to be weak enough perturbations
so the analysis of Ref. [49] is justified.
What we do learn from the overlap expansion result (3.18) is that the non-orthogonality
is a much smaller problem on more open lattices which involve large loops. While the
honeycomb is a good candidate on this score, to put the issue beyond doubt we now consider
a decorated version of the lattice.
3.3.3
Decoration scheme
In this section, we propose a modification to our earlier arguments which makes the overlap
expansion essentially exact. Consider the decorated honeycomb lattice shown in Fig. 3.8
where we insert N (an even integer) sites between neighboring sites of the usual honeycomb
lattice. The dimer structure of this lattice, including the number of dimer states, is exactly
the same as before except that having a dimer between sites 1 and 2 corresponds to a chain
of (N + 2)/2 dimers beginning at site 1 and ending at site 2. Not having a dimer between
sites 1 and 2 corresponds to having a chain of N/2 dimers beginning at site b1 and ending at
site a2 . The Klein Hamiltonian is correspondingly modified by including Klein projectors
for the added sites.
Majumdar and Ghosh [46] showed that the valence bond state is the only ground state
of the Klein Hamiltonian for a one-dimensional spin chain with an even number of spins.
Therefore, the conclusions regarding the Klein model on the honeycomb lattice (linear
independence of valence bond states, valence bond states span the ground state manifold,
etc) carry over directly to the decorated honeycomb lattice.
While decorating does not introduce any new technical problems, there is a significant
technical advantage with respect to the overlap expansion. The smallest two loops on the
hexagonal lattice are length 6 and 10 from which we obtained that the relative orders of
the leading and error terms in the overlap expansion were x6 and x10 . The smallest loops
51
on the decorated hexagonal lattice have lengths 6(N + 1) and 10(N + 1). Repeating the
previous analysis, we will find that the leading and error terms in the overlap expansion
are x6(N +1) and x10(N +1) . The ratio of error term to leading term has improved from x4 to
x4(N +1) . In the large N limit, the error term is “rigorously” negligible but we propose that
even fairly small values of N may suffice to capture the qualitative features of the large N
limit.
While we have added complexity to the lattice, we do not have to increase the order of
the spin interaction. Consider the following as a perturbation to the decorated honeycomb
lattice Klein model:
δH
= J
si · sj +
ij
v
(
s1 · sb1 )(
s3 · sb3 )(
s5 · sb5 )
+(
s1 · sa1 )(
s3 · sa3 )(
s5 · sa5 )
(3.19)
The first term is a nearest neighbor interaction over all spins while the second term is over
all elementary plaquettes, such as the one in Fig. 3.8. A 6 spin interaction is sufficient, even
though we have many more spins in the loop, because having a (1b1 ) bond automatically
implies the other bonds in the chain connecting 1 and 2. Our previous analysis carries over
to the present case and we conclude:
Hαβ = −Jx6(N +1) αβ + vnf l,α δαβ
+ O(vx6(N +1) + Jx10(N +1) )
= −tαβ + vnf l,α δαβ + O(vx6(N +1) + tx4N )
(3.20)
where t = Jx6(N +1) and otherwise the notation is the same.
Clearly, by decorating enough we can make the matrix elements beyond the dimer model
arbitarily small and thus realize the physics, including Cantor deconfinement, present in
generic, weak perturbations of the honeycomb QDM.
52
Spinons
In the highly decorated limit, one may show that nearest-neighbor valence bond states are
separated by a finite gap from the excited states of the Klein model. In this limit, we are
connecting a set of Majumdar-Ghosh[46] (MG) chains into a two-dimensional network. We
may describe the low energy excited states of our system in terms of the well studied spinon
defects of the MG chains, which are widely believed to be gapped[69, 70, 71]. Here we give
an outline of our argument and relegate technical details to Appendix E.
In Appendix E, we consider what happens when we put these chains together for different
values of a tunable parameter in our model: the ratio of Klein scales (the coefficient Ci in
Eq. 3.5) of the Klein projectors of the decorated and reference sites. For large values
of this ratio, the excited states are represented by “microscopic” spinons localized on the
reference sites. For small values of the Klein ratio, the excited states are extended and may
be interpreted as MG spinons scattering at the vertices. There is a first order transition
between these limits. In both limits, there is a gap between the VB manifold and the spinon
states, as depicted in Fig. 3.9.
Spinons are the natural excited states (outside of the VB manifold) in the high decoration
limit. For an unperturbed Klein model, the VB manifold is degenerate so these excitations
are mobile. The next question is what happens when the degeneracy of the ground state
manifold is lifted. We argue that this has a small but vital effect on the spinon dynamics.
At the RK point and in liquid phases, we expect the spinons to be deconfined. In the
crystalline phases, we may consider a pair of test spinons, holding one member fixed and
considering the quantum mechanics of the other. If the wavefunction of the non-fixed spinon
has spatial extent L, this would have an energy cost of order c L2 where c is the energy cost
per unit area of scrambling the crystalline background. The L-dependence of the kinetic
energy varies as 1/L2 . The implication is that while c is a much smaller scale than the
spinon gap, spinons moving in a crystalline background are confined at sufficiently long
length scales.
53
3.3.4
Square lattice
The square lattice is another two-dimensional bipartite lattice for which (nearest-neighbor)
valence bond states are linearly independent[56]. We may orient the bonds on the square
lattice so that two states differing by a (minimal) length 4 loop, have positive overlap. The
problem with applying our approach to the bare square lattice is that the Klein model
has ground states outside of the valence bond manifold (Fig. 3.10). These extra states do
not arise when we consider the decorated square lattice. Consider perturbing a decorated
square lattice Klein model with:
δH = J
si · sj +
ij
v
(
s1 · sb1 )(
s3 · sb3 ) + (
s1 · sa1 )(
s3 · sa3 )
(3.21)
where the first term is a nearest neighbor interaction and the second sum is over all elementary plaquettes, the spins 1234 labelling the 4 sites of a square plaquette in clockwise order
and, as before (see Fig. 3.8), the labels ai and bi denoting the first counterclockwise and
first clockwise neighbor of spin i. By arguments similar to the honeycomb case, we obtain
an effective Hamiltonian:
Hαβ = −Jx4(N +1) αβ + vnf l,α δαβ
+ O(vx4(N +1) + Jx6(N +1) )
= −tαβ + vnf l,α δαβ + O(vx4(N +1) + tx2N )
(3.22)
Here ij is a matrix that is 1 if states |i and |j differ by the (minimal) length 4 loop
and zero otherwise. Therefore, we realize the physics of the square lattice QDM. Note
that without the decoration, the error would be order x2 = (1/2), as opposed to the bare
honeycomb case where the error is order x4 = (1/4). We also emphasize that these “small”
terms multiply matrices so in the undecorated case, the overlap expansion is not obviously
justified.
54
3.4
Other Valence Bond Phases in d = 2 and d = 3
We now sketch the application of our strategy to obtain the rest of dimer model physics,
including RVB phases. The points to be made concern the choice of lattices and phase
conventions.
3.4.1
Non-bipartite lattices in d = 2
The simplest d = 2 non-bipartite lattice is the triangular lattice. Numerical evidence
suggests that (nearest-neighbor) valence bond states are linearly independent[65]. As with
the square and honeycomb lattices, we may orient bonds so that states differing by a
(minimal) length 4 loop, have positive overlap.[72] As with the square lattice, the Klein
model admits non-valence bond ground states, though the problem is more serious with the
triangular lattice (see Fig. 3.11). Decoration eliminates these possibilities by removing the
triangular nearest-neighbor structures. Applying our strategy to the decorated triangular
lattice allows us to reproduce the physics of the triangular lattice QDM, including its RVB
phase[51]. By calculations similar to Appendix E, one may show that spinon excitations are
gapped. For the triangular lattice, it is known that collective excitations within the valence
bond manifold are also gapped[51, 73], a conclusion which will remain valid for the decorated
case. Therefore, we have constructed a model that shows a stable, SU(2)-invariant RVB
liquid phase.
Another non-bipartite lattice is the pentagonal lattice[74] shown in Fig. 3.12. There
are currently no formal proofs for the pentagonal lattice regarding the issues of linear
independence of (nearest-neighbor) valence bond states and whether the set of these states
spans the ground state space of the corresponding Klein model. However, it was explained
in Ref. [56] that the most important ingredients of their proofs for the honeycomb lattice
are its relatively low coordination number (3); relatively large minimum loop size (6); and
the absence of triangular structures in the lattice. The pentagonal lattice, has sites of
coordination 3 and 4, minimum loop size 8, and no triangular structures, suggesting that
the arguments may be adapted to this lattice. As before, it is possible to orient bonds so
55
that the overlap of states differing by a (minimal) length 8 loop always has the same sign.
While in the square, triangle, and honeycomb cases, the sign of the overlap is a matter of
convention (which we chose as positive), for the pentagonal lattice, only the negative sign is
possible. In fact, in Appendix D, it is shown that using the fermionic convention, one may
always obtain the negative sign independent of lattice details. Therefore, to generate the
QDM kinetic energy, we must perturb the Klein model with a ferromagnetic interaction.
From a perturbation of the form:
δH
= −J
si · sj +
ij
v
(
s1 · sb1 )(
s3 · sb3 )(
s5 · sb5 )(
s7 · sb7 )
+(
s1 · sa1 )(
s3 · sa3 )(
s5 · sa5 )(
s7 · sa7 )
(3.23)
where the first term is over all nearest-neighbor spins and the second term is over elementary
plaquettes (ai and bi , once again, denoting the first counterclockwise and first clockwise
neighbors of spin i), we may obtain the quantum dimer Hamiltonian for the decorated
pentagonal lattice:
Hαβ = −tαβ + vnf l,α δαβ + O(vx8(N +1) + tx2N )
(3.24)
Here ij is a matrix that is 1 if states |i and |j differ by the (minimal) length 8 loop
and zero otherwise. Therefore, we realize the physics of the pentagonal lattice QDM. We
have checked that in the classical limit, the dimer-dimer correlations decay exponentially
and in Appendix F, we present numerical evidence that monomers are deconfined. As both
features also hold at the RK point, we may repeat the arguments described in Ref. [51]
for the triangular lattice to conclude that the pentagonal lattice QDM also shows an RVB
liquid phase, a result which may be transcribed into spin language as discussed above.
56
3.4.2
Non-bipartite lattices in d = 3
The face-centered cubic (FCC) is a three-dimensional non-bipartite Bravais lattice with each
site having 12 nearest-neighbors. The undecorated lattice has triangular structures involving
two neighboring facial sites and the two corners which are their common neighbors. This
will lead to non-valence bond ground states in the FCC Klein model. Decoration eliminates
the triangular structure and hence this type of pathology. The shortest resonance loops
are length 4. In the fermion sign convention, these loops come with negative sign. A
perturbation consisting of a ferromagnetic exchange and 4-spin resonance interaction will
reproduce the QDM results for the decorated lattice. We expect the resulting model to
show a Z2 RVB phase near its RK point[52].
3.4.3
Bipartite lattices in d = 3
For the diamond lattice we pursue the same strategy as above. The properties of the
diamond lattice we require are the following.
The diamond lattice is bipartite, has coordination four, and the shortest resonance loops
are of length six. The Klein Hamiltonian again has nearest-neighbour dimer coverings as
ground states, although, as for the case of the pentagonal lattice, no theorem exists excluding
other ground states. It is likely that extra states may be excluded by decorating the lattice.
The number of dimer ground states, ngs , is exponentially large in the number of sites, N ,
but it is not known exactly. Defining the ground state entropy per site as S = (1/N ) ln ngs ,
an accurate series expansion by Nagle[29] yields S ≈ 0.265.
We now try to mimic an RK quantum dimer model for the diamond lattice. As for the
case of the honeycomb lattice, we do this by adding a nearest-neighbour exchange term to
induce a kinetic term and in addition, a ring term to generate a potential term. We then
expect the resulting model to exhibit, near the effective RK point, a U(1) RVB liquid phase
with algebraically decaying correlations as well as gapless photonic gauge excitations, as
discussed in detail in Ref. [53].
This liquid phase will give way, upon making the potential term more attractive, to
57
a columnar-type solid. For an increasingly repulsive potential, the scenario of Cantor deconfinement predicted for the two-dimensional case is simplified. Technically, there are no
relevant lock-in terms in three dimensions so that the deconfined region simply acquires an
increasing amount of U(1) flux as v/t is increased through the RK point; finally, a staggered
solid, with the maximal amount of U(1) flux allowed by microscopic constraints,[52, 53, 55]
is reached. We cannot say whether this will happen continuously or via a first order transition.
3.5
Dynamical selection of gauge structures: pyrochlore lattice
We construct a Klein model with Z2 order and a Kivelson-Klein model with U (1) order
which takes advantage of the bipartiteness of the dual lattice. This nicely illustrates the
dynamical selection of the low-energy gauge structure present in topological phases.
3.5.1
The Klein model
The undecorated pyrochlore lattice (Fig. 3.13) does not lend itself straightforwardly as a
starting point for dimer models obtained via the Klein route because its basic building
block, the tetrahedron, supports more dimer coverings than linearly independent singlet
states. By sufficiently decorating the lattice, the orthogonality problem is solved. The
shortest resonance loops are length 6 and the fermionic convention may be used to make
the minimal overlaps come with negative sign. Perturbing the decorated pyrochlore Klein
model with a ferromagnetic nearest-neighbor interaction and 6-spin ring interaction, we
obtain an effective Hamiltonian mimicking the pyrochlore lattice QDM, which includes a
Z2 RVB phase.
58
3.5.2
The Kivelson-Klein model
A modified version of the Klein Hamiltonian, which we will refer to as the Kivelson-Klein
Hamiltonian, may be used to produce a model displaying a U (1) RVB phase. [75] Its
Hamiltonian is of the same form as Eq. 3.3 but the definition of N (i) is changed. The
projection now acts not on a site and its nearest neighbours but instead on the four sites
of a tetrahedron:
HKK =
P̂tet .
(3.25)
tet
We note that the simple decoration trick described above cannot as usefully be applied to
the Kivelson-Klein model, as here increasing the number of sites in the tetrahedron does
not lead to an increase in the number of dimers required.
Ground states of the Kivelson-Klein model
Evidently, each state in which each tetrahedron contains at least one singlet bond is a ground
state of the above Hamiltonian, Eq. 3.25. How can this be related to dimer coverings of the
pyrochlore lattice?
First, note that (i) the number of tetrahedra equals twice the number of sites, Nt = 2N ,
and that (ii) the number of hardcore dimers, Nd ≤ N/2, as a dimer involves two sites.
From this it follows that Nd ≤ Nt , the equality sign holding for hardcore dimer coverings.
However, the requirement of having at least one dimer in each tetrahedron gives (iii) Nd ≥
Nt . We therefore see that (ii) and (iii) imply that those hardcore dimer coverings of the
pyrochlore lattice for which each tetrahedron contains exactly one dimer are ground states
of the Kivelson-Klein Hamiltonian.
The ensemble of these states maps onto the ground states of the six-vertex model on
the diamond lattice, or equivalently onto the ground states of the pyrochlore Ising antiferromagnet. This can be seen as follows. First, note that the lattice of tetrahedra defined by
the pyrochlore lattice is the bipartite diamond lattice. One diamond sublattice sits at the
centre of the ‘up’ tetrahedra, the other one at the centres of the ‘down’ tetrahedra. Now let
us define the Ising spins on the pyrochlore lattice as follows. For an up (down) tetrahedra,
59
the pair of spins at the two ends of a dimer point up (down), and the other pair points
down (up). This defines a one-to-one mapping of dimer to Ising states; crucially, on each
tetrahedron, two spins point up and two point down, thus putting the tetrahedron into an
Ising ground state. (The mapping to the six-vertex model on the diamond lattice proceeds
by calling an up (down) spin an arrow pointing from the centre of an up (down) tetrahedron
to a down (up) tetrahedron).
Ground state correlations
The total number of ground states (assuming that there are none in addition to the abovementioned dimer states) gives rise to an extensive ground state entropy well-approximated
by the Pauling entropy SP auling = (1/2) ln(3/2).
Using the mapping to an Ising magnet, it is straightforward to calculate the correlator
between singlet bonds averaged over the ground state manifold. To do this, note that each
of the six dimer positions on a bond of a given tetrahedron corresponds to an Ising ground
state of that tetrahedron. In turn, the corresponding vertex of the six-vertex model describes
a net flux, the direction of which is given as follows. Consider a cube circumscribing the
tetrahedron in question, so that the bonds of the tetrahedron are face diagonals of the
circumscribing cube. The direction of the flux (i.e., the average direction of the four arrows
of the given vertex) now points from the centre of the cube through the midpoint of the
face of which the bond occupied by the dimer resides.
Using the theory developed in Ref. [52], one can read off that the dimer correlations are
simply dipolar. Briefly, this follows from the observation that, upon coarse graining, the
smaller the coarse-grained flux, the more microstates (prior to coarse-graining) correspond
it. Modelling this by an effective quadratic weight on the flux configurations leads to simple
magnetostatics.
For example, the connected correlator between a pair of dimers located on the top of an
up tetrahedra, separated by a vector r which makes angle θ with the z axis, is proportional
to the dipolar form (3 cos θ 2 − 1)/r 3 .
60
Finally, this model can again in principle be “Rokhsar-Kivelsonized”, i.e. by adding
appropriate perturbations to Eq. 3.25, we may generate an effective Hamiltonian which
acts on the Kivelson-Klein ground state manifold in a manner similar to Eq. 3.2 on the
space of dimer coverings. We do not do this here for a number of reasons. Firstly, the
expected phase diagram has the same topology as that discussed for the diamond lattice in
the previous section, so no new phases are obtained. Secondly, the dimer dynamics is rather
messy. The shortest resonance loop now involves six dimers straddling a hexagonal loop
of the pyrochlore lattice – and there are several symmetry-inequivalent loops of this type.
(This is reminiscent of the – exactly soluble – kagome dimer model proposed in Ref. [76].) In
addition, for a simple perturbing nearest-neighbour exchange, the resonance term vanishes
in the leading order of the overlap expansion, so that a more complex perturbation is needed.
3.6
Discussion and outlook
We have presented spin-1/2 Heisenberg Hamiltonians that realize a large class of valence
bond phases. In particular they realize Z2 RVB phases in d = 2 and d = 3, the U (1) RVB
phase in d = 3 and the Cantor deconfined region in d = 2. These phases have previously
been shown to exist in quantum dimer models with dimers standing in for valence bonds. In
this paper we have constructed Klein models that exhibit ground state manifolds spanned by
nearest neighbor valence bond states and then perturbed them to to realize quantum dimer
models within these manifolds. This perturbation is done within the framework provided
by the overlap expansion, made arbitrarily accurate by a decoration procedure that we have
introduced.
The decoration has the effect of expanding the length scale on which the Hamiltonian
acts directly. However in order to stabilize the phases in quantum dimer models, we do
not need to go to infinite decoration – it would be enough to suppress subleading terms
sufficiently. In this fashion we obtain spin models with interactions of finite range[82]. While
large decorations are needed to realize the simplest dimer models under analytic control,
there is little reason to doubt that the extent of decoration could be reduced drastically,
61
and even eliminated altogether on some of the lattices without sacrificing the various phases
of interest. The subleading interactions will not necessarily uniformly tend to subvert such
phases and it is also possible to add other terms that would stabilize them. Showing how
this can work is an obvious task for the future, as is the construction of mathematically
rigorous proofs for various statements in this paper that are made by appealing to a small
parameter. One promising approach is that of Ref. [77], which appeared at the same time
as our work, where the phases of the d = 2 bipartite quantum dimer model are realized by
perturbing a Klein model with ring exchange terms. We note that the RVB phase on the
triangular lattice is generically stable to small perturbations, including (perhaps) the error
term resulting from a finite decoration, because it is gapped.
We emphasize that our central result has been the demonstration that spin liquid phases
can be realized in SU(2)-invariant models. However, the actual models we have given involve rather complicated geometries and Hamiltonians without direct experimental relevance. While it may be possible to engineer highly decorated lattices, a more important
task perhaps, now that the question of principle is settled, is to refocus on studying much
simpler Hamiltonians. Our insistence on a specified form of the wavefunction (containing nearest neighbor valence bonds alone) has led to fairly complicated Hamiltonians but
simpler Hamiltonians can exhibit the same phases with more elaborate ground state wavefunctions. Indeed, the situation for the simplest lattice under consideration in this paper,
the (undecorated) honeycomb lattice, looks quite promising. Previous exact diagonalisations of a J1 − J2 − J3 Heisenberg model on this lattice[78] have clearly demonstrated the
existence of the staggered VBS. The current data appears not inconsistent with a scenario
in which the magnet leaves the staggered phase but never reaches the fluxless plaquette
phase. We are optimistic that it will be possible to realize the physics discussed in this
chapter in such simpler models.
62
Figure 3.3: The kinetic (potential) energy operator of the simplest quantum dimer model
on a given lattice will flip (count) the dimers on the simplest flippable plaquette. The
basic resonance move for the honeycomb lattice and the three basic moves on the triangular
lattice are shown. The construction may be obviously generalized to arbitrary lattices and
dimensions.
63
Figure 3.4: Phase diagram for a bipartite QDM in d=2 from Fradkin et. al. [55].
i
i
-i
i
i
i
RVB
columnar
12x 12
columnar
0 columnar/plaquette
staggered
1 staggered
v/t
Figure 3.5: Phase diagram for a non-bipartite QDM in d=2 from Moessner and Sondhi[51].
64
Figure 3.6: Transition graph construction for two valence bond coverings of the square
lattice (the construction for other lattices is similar). The dots are the lattice sites and
the thick lines denote that the two sites form a singlet bond. The singlet orientation may
be specified, for example, by having the bonds point from the gray sites to the white
sites. The transition graph is formed by overlaying the two configurations resulting in
a graph (lower) containing double bonds and closed loops of varying (even) lengths. In
the above example, there are two double bonds and three loops of lengths 4, 6, and 16.
The magnitude
the two valence bond coverings is then given by
√ of the overlap between
1
. Thus, while the overlap between two arbitrary
|S| = 23 (1/ 2)4+6+16 = ( 12 )10 = 1024
valence bond coverings is never zero, it is usually a small number. This is the basis for the
overlap expansion discussed in the text.
65
1
6
2
5
3
4
Figure 3.7: The elementary plaquette of the honeycomb lattice.
66
a1
1
b6
a6
b5
b1
a2
6
2
5
3
a5
a3
b3
4
b4
b2
a4
Figure 3.8: The decorated honeycomb lattice where N (an even integer) two-fold sites are
inserted between the old sites. This drawing shows N = 4. The labels a1 and b1 designate
the first counterclockwise and first clockwise neighbor of spin 1 where clockwise is with
respect to the loop 123456. For the undecorated case, a1 and b1 are just sites 6 and 2.
67
E
MG spinon continuum
MG spinon continuum
Micro-spinon band
VB excitations
QDM ground state
VB excitations
QDM ground state
Figure 3.9: A cartoon of the spectra for the limiting cases where the Klein ratio is large
(left) and small (right). In the small Klein ratio case (right), the lowest spin excitations
are described by extended spinon wavefunctions. These scattering states are present even
in the large ratio case (left) but here the lowest excitations, which we call “microscopic”
spinons, are described by a wavefunction having peaks at the reference sites and decaying
on the chains. The decay rate can be made arbitrarily fast by tuning the Klein ratio. The
bandwidth of these localized states depends on the decoration and is zero in the infinite
decoration limit. Whether or not the valence bond excitations are gapped depends on the
details of the specific QDM. For example, the topological phase on the triangular lattice is
gapped while the RK point on bipartite lattices have gapless excitations. Also, the valence
bond excitations will not, in general, have a finite bandwidth; the figure is drawn this way
to emphasize the spin gap. Details are given in Appendix E.
68
Figure 3.10: The Klein model for the square lattice, with periodic boundary conditions,
permits ground states which are not in manifold spanned by nearest-neighbor valence bond
states such as this one. Here the thin lines form the lattice and the thick lines denote singlet
pairings. Note that the Klein condition is satisfied at every lattice site.
Figure 3.11: The Klein model for the triangular lattice admits many nontrivial non-dimer
ground states, such as this one. The thin lines show the lattice and the thick lines denote
singlet pairings. The dots represent free spins. Note that the Klein condition is satisfied at
every lattice site.
69
Figure 3.12: The pentagonal (Sutherland-Shastry) lattice. It is the dual of the ShastrySutherland lattice, which is indicated by the dashed lines.
Figure 3.13: The pyrochlore lattice, a network of corner-sharing tetrahedra
Appendix A
An overview of height
representation theory
A.1
The height representation
The height representation provides a natural way to construct field theories of dimer models
on bipartite lattices. While our primary concern is the two-dimensional honeycomb lattice,
the height mapping applies to any bipartite lattice and may be generalized to arbitrary
dimension[52]. The strategy is to find a free energy functional which satisfies the basic
symmetries of the microscopic system and reproduces the important long distance characteristics, such as the decay of dimer-dimer correlations. We then use this functional to
consider questions not easily addressed using the microscopic model, such as the nature of
the interaction of certain excitations or the stability of the model to generic perturbations.
A dimer covering of the honeycomb lattice may be represented by assigning a height
variable hi to each site i of the triangular lattice dual to the hexagonal lattice on which
the dimers lie. This dual lattice is formed by the centers of the hexagons. The heights are
assigned as follows. Assign a value of zero height to some arbitrary site on the triangular
lattice. Moving from this site to a nearest neighbor site by moving clockwise around an up(down-) triangle, increase (decrease) the height by +2 (-2) if a dimer is crossed. If a dimer
70
71
is not crossed, then decrease (increase) the height by -1 (+1). According to these rules,
traversing a closed loop counter-clockwise in the dual lattice will result in a height difference
of 0 if the constraint of one dimer per site is satisfied everywhere inside. Deviations from
this constraint (monomers or multiple dimers on a site) manifest as vortices in the height
variable. For example, the height difference is +3 (-3) if a single positive (negative) monomer
is enclosed by the loop. Examples of dimer coverings and the associated height mappings
are given in Fig. A.1.
In a coarse-grained description, the hi are replaced by a real, continuum field h(x).
We now consider two examples of how this representation may be used to construct field
theories of dimer models and some applications relevant to this thesis.
A.2
A.2.1
Application to spin ice
Classical dimers on the honeycomb lattice
The tilt of a configuration is the average value of the gradient of its height field in a given
direction, for example:
1
L
|∇x h|dx. In the absence of potentials, the states with small (or
zero) tilt will be favored in the free energy for entropy reasons (i.e. there are more such
states). This motivates the following guess for the free energy (which is entropic):
F {h(x})
= −S{h(x)} =
T
K
dx
2
|∇h|2 + V (h(x))
(A.1)
The gradient term penalizes tilted states and the constant K will be determined below.
The form of the potential energy may be determined by noting that the physics should not
depend on which site we choose as our zero height reference point. Choosing a different
origin is equivalent to uniformly shifting the heights: h(x) → h(x) + c where c is some
integer. In our model, the minimum such shift is one unit so the potential V must be a
periodic function of h with period one. A common simplification is to approximate V by
its lowest frequency component:
V (h((x)) ≈ λ cos(2πh)
(A.2)
72
If λ < 0, this term favors configurations where the height function takes integer values
everywhere and when combined with the first term, would favor the system locking into
smooth, flat configurations. As discussed in Ref. [32], the actual outcome depends on the
value of K and it turns out that this phenomenological model also exhibits a rough phase
where the potential term is irrelevant (in the sense of the renormalization group). The
spin ice [111] magnetization plateau is described by this rough phase so we will neglect the
potential term in the following.
The next step is to choose the constant K so that our effective theory reproduces
the long distance behavior of the microscopic classical dimer problem. For the honeycomb
lattice, this problem has an exact solution and the long distance behavior of the dimer-dimer
correlation function where both dimers are oriented vertically (labelled the “1” direction
below; see Fig. A.1) is:
c11 (x) = n1 (x)n1 (0) =
1 1
4πx
)
−
cos(2θ)
cos(
2π 2 r 2
3
(A.3)
where r and θ are the usual polar coordinates and n1 (x) is 1 if there is a vertical dimer
on the link to the right of position x (which labels a site on the dual height lattice) and 0
otherwise. In terms of heights, the density of dimers having orientation κ is given by nκ =
1−(êκ ·∇lat )h /3 where ∇lat is the lattice gradient and êκ is a unit vector perpendicular to
the dimer orientation κ. From this, we may guess the corresponding form in the continuum
case:
nκ (x) =
1 − (êκ · ∇)h
+ fκ (h(x), x)
3
(A.4)
where fκ (h(x), x) is a function whose form we will now determine. While naively, we may
expect just the first term to be the correct continuum expression, it turns out (as will be
shown below) that this term is insufficient in reproducing the behavior given in Eq. A.3.
Because nκ is a local quantity, we expect fκ (h(x), x) will depend only on the local arrangement of dimers which translates into a local distribution of height differences. Suppose
a particular dimer pattern occurs in some part of the lattice and the same dimer pattern
occurs elsewhere. Then the value of nκ should be the same in both regions. While by
73
construction, both regions will have the same distributions of height differences, the values
of the heights themselves will in general be uniformly shifted from region to region. An
example of this is given in Fig. A.2 which indicates that for the honeycomb (square) lattice, this uniform shift is 3 (4). This implies that for the honeycomb lattice, the function
fκ (h(x), x) should be a periodic function of h(x) with period 3.
The simplest form of such a function is:
fκ (h(x), x) = gκ (x)e
2πih(x)
3
+ c.c.
(A.5)
More generally, we may include terms for the higher wavevectors in height space that also
2π
have period 3 ( 4π
3 , 2π, etc.) but the lowest one ( 3 ) will be the most relevant in terms of the
long distance behavior. The function gκ (x) is chosen in order to match Eq. A.3. It turns out
that gκ (x) = λκ e4πix/3 where λκ is a constant and x refers to the horizontal coordinate (not
|x|). This is related to the fact that the macroscopically flat states will have a microscopic
height fluctuation with a characteristic wavevector of 1 12 lattice constants (see Fig.A.1).
With this conjectured form for the dimer density, we may compute the dimer-dimer
correlation function for vertical (κ = 1) dimers. We may proceed using the familiar path
integral approach. Our continuum theory implies a partition function given by:
Z[J] =
D[h]e−S[h;J]
(A.6)
where the integral is over all possible height configurations; J(x) is a source term; and the
action is given by:
S[h; J] =
1
2
1
1
d2 q 2 Kq
h
h
+
J
h
+
J
h
q −q
q q
−q −q
(2π)2
Kq 2
Kq 2
(A.7)
where hq denotes the (2d) Fourier transform of h(x) and so on. Performing the Gaussian
integral, we obtain:
Z[J] = Z[0] exp
1 2
d2 q Jq J−q (2π)2 Kq 2
(A.8)
from which we may conclude:
hq1 h−q2 =
δ2 Z
1
=
(2π)2 δ(q1 − q2 )
J=0
δJq1 δJ−q2
Kq12
(A.9)
74
and:
1
(h(x) − h(0))2 = (h(0)2 − h(x)h(0))
2
d2 q1 d2 q2
(1 − eiq1 ·x )hq1 h−q2 =
(2π)2 (2π)2
d2 q (1 − cos q · x)
=
(2π)2
Kq 2
r
1
ln( )
≈
2πK
a
(A.10)
where a << r is a length scale (of order the lattice spacing) associated with regularizing
the potential at short distances. This information may be used to obtain the dimer-dimer
correlator:
n1 (x)n1 (0) =
2πi
1 1
+ ∂x h(x)∂x h(0) + λ2 e4πix/3 e 3 (h(x)−h(0) + c.c. (A.11)
9 9
Each of the two averages may be computed using the above results. We obtain for the first
term:
∂x h(x)∂x h(0) =
=
=
=
=
This expression, choosing K =
π
9,
d2 q1 d2 q2 iq1 ·x 2
e
qx1 hq1 h−q2 (2π)2 (2π)2
d2 q qx2 eiq·x
(2π)2 Kq 2
d2
d2 q 1 − cos q · x
dx2
(2π)2
Kq 2
r d2 1
ln( )
2
dx 2πK
a
1 y 2 − x2
2πK r 4
(A.12)
reproduces the second term of Eq. A.3. We obtain the
other term by using Eq. A.8 with Jq = (eiq·r − 1).
e
2πi
(h(x)−h(0))
3
With K = π/9 and λ2 =
1
4π 2 a2 ,
2πi )2
d2 q 1 − cos q · r 3
(2π)2
Kq 2
4π 2 1
r ln( )
= exp −
9 2πK
a
r − 2π2
9K
=
a
= exp (
we obtain Eq. A.3.
(A.13)
75
A.2.2
Interaction between defects in spin ice
As mentioned in the main text, spin ice on the [111] plateau is a layered system, each layer
being described by a classical dimer model. A string defect involves creating a pair of defects
in each layer with an added interlayer constraint. The defects are sites of the honeycomb
lattice having two dimers; the two defects are on different sublattices so may be labelled as
“positive” and “negative”. In the continuum theory, we consider a defect density function
σ(x) and the multivaluedness of the height field is expressed in the equation:
C
=3
∇h · dr
d2 rσ(
r)
(A.14)
S
where S is the area enclosed by the loop C. We may proceed by analogy with the 2d XY
model and divide h into “dimer” (spin-wave) and “defect” (vortex) contributions. We follow
the presentation of Ref.[31].
h(x) = hsw (x) + hv (x)
(A.15)
∇ · ∇hv = 0
(A.16)
∇ × ∇hsw = 0
(A.17)
where
∂hv
∂ h̃v
∂ h̃v
2
v
It is useful to define the field h̃v such that ( ∂h
∂x , − ∂y ) = ( ∂y , − ∂x ). Notice that |∇hv | =
|∇h̃v |2 and Eq. A.14 may be rewritten (using Stokes’s theorem) as:
∇2 h̃v (x) = 3σ(x)
(A.18)
The equation may be solved using Green’s functions:
h̃v (x) =
3
2π
dyσ(y) ln(
|x − y|
)
a
Using the above information, we may rewrite the Hamiltonian for the system as:
F {h(x})
T
K
dx|∇h|2
2
K
=
dx(|∇hsw |2 + |∇hv |2 )
2
F {h(x}) K
+
dx|∇h̃v |2
=
sw
T
2
=
(A.19)
76
F {h(x}) K
−
dxh̃v ∇2 h̃v
=
sw
T
2
F {h(x}) 9K
|x − y| =
+
dxdyσ(x)σ(y) − ln
sw
T
4π
a
F {h(x}) 1
|x − y| =
+
dxdyσ(x)σ(y) −κ ln
sw
T
2
a
(A.20)
where κ = 1/2. This shows that the interaction between two defects separated by distance
r is given by p1 p2 V (|r
1 − r
2 |) where pi is +1 (-1) for a positive (negative) defect and
V (R) = −κ ln(R/a).
A.2.3
Summary
To summarize, we provided arguments for why the free energy for classical dimers on the
honeycomb lattice is given by:
K
F {h(x})
=
T
2
where K =
π
9
dx|∇h|2
(A.21)
for the honeycomb lattice. If we were working on a lattice where we did not
know the exact solution, then this constant K would need to be determined numerically by
matching the continuum dimer-dimer correlator with numerical simulation of the correlator
of the microscopic model. We then used this to determine the coupling constant of the
interaction between defects which was required in the analysis of string defects in spin ice
discussed in the main text. We emphasize that this interaction between defects is entropic
in origin. It is instructive to phrase this in terms of the microscopic model. If two defects
are fixed at positions r1 and r2 and everywhere else, the one dimer per site rule is obeyed,
we may ask how many dimer coverings are consistent with the constraint and how this
number changes as the defects are moved. If moving the defects increases the number of
dimer coverings of the remaining sites, then the defects will move in order to maximize the
dimer entropy. What the above analysis yielded is the nontrivial result that the number of
dimer coverings scales with defect separation r as r −1/2 at long distances.
77
A.3
A.3.1
Application to quantum dimer models
Quantum dimers on bipartite lattices
The height representation provides a route to constructing a field theory for quantum dimers
moving under a quantum dimer Hamiltonian. The action conjectured by Henley[25] for this
problem is sometimes called the quantum Lifshitz model:
S[h] =
1
d2 x
1
1
(∂τ h)2 + ρ2 (∇h)2 + ρ4 (∇2 h)2 + λ cos(2πh)
2
2
2
(A.22)
The coefficient −ρ2 = (v/t)−1 where t and v are the coefficients of the kinetic and potential
energy terms in the RK quantum dimer Hamiltonian. The coefficient ρ4 depends on the
lattice (for the honeycomb, ρ4 = (π/18)2 ) and, similar to the classical case discussed above,
is chosen to correctly recover the dimer-dimer correlations. A special property of the RK
point (where v = t) is that the correlations are the same as in the classical model so the
procedure is literally the same as what we used above to calculate K, except the action is
now described by Eq. A.22.
Eq. A.22 is the simplest action which has a number of features which are required
to discuss the quantum dimer model but is not easily deduced by deductive argument.
Moessner et.al. [50] motivated this form by noting that the gradient-squared term (with
ρ2 > 0) and the cosine term (with λ < 0) favor the flat states (columnar and plaquette
phases). If ρ2 < 0, then the quadratic term favors the tilted state (staggered phase).
At the RK point itself, quadratic term disappears, the cosine term is irrelevant, and the
remaining Laplacian squared term does not distinguish between the flat and tilted states.
This qualitatively describes the bipartite QDM phase diagram.
A.3.2
Interaction of defects
As in the classical case, monomer defects may be described as vortices in the height field
and Eqs. A.14 and A.18 will still apply. We may use this to rewrite the action at the RK
point for the case of time independent correlations:
S[h] =
1
d2 x ρ4 (∇2 h)2
2
78
=
=
9
d2 x ρ4 σ(x)2
2
9ρ4
dxdyσ(x)σ(y)δ(x − y)
2
(A.23)
where we have dropped the irrelevant cosine term. In contrast to Eq. A.20, we find that
monomer defects at the RK point do not interact with one another beyond having hard
cores. Therefore, we obtain that monomer defects are deconfined at the RK point.
A.3.3
Stability of the RK point
Fradkin et. al. [55] considered the influence of generic perturbations on the action (A.22)
at the RK point. We will summarize the results of their work of particular importance to
this thesis. The first observation was that:
√
√
1
1
3
3
∂y h)( ∂x h +
∂y h)
S3 = g3 (∂x h)( ∂x h −
2
2
2
2
(A.24)
is a relevant perturbation that will, when g3 is nonzero and negative, drive the system into
a tilted phase pointing along one of the three dimer orientations of the honeycomb lattice.
Even if g3 were strictly tuned to zero, they proposed higher order terms which could also
break the degeneracy of tilts which is a feature of the RK point. Assuming the system was
in a tilted phase to the right of the RK point, they then considered the influence of the
most general perturbation allowed by the microscopic model:
V (h(x), x) =
VG eiGh +Gx ·x
(A.25)
G
where G = (Gh , Gx ) where Gx is a reciprocal vector of the honeycomb lattice and Gh is a
reciprocal vector of the height lattice. They found that if the tilts were commensurate with
the lattice, then the term in the above sum with the wavevector of the commensuration was
a relevant perturbation. In this case, the system would lock into a confining crystalline phase
with gapped excitations. If the tilt was incommensurate with the lattice, then the system
would be in a crystalline phase with gapless excitations that were logarithmically confined.
The magnitude of the commensurate wavevector is inversely proportional to the magnitude
of the tilt so as the tilt is made very small, by tuning g3 for example, then the commensurate
79
states would occur less and less often. In addition, the gap in the commensurate phase was
estimated to be exponentially small at small tilts.
The result is that tuning a generic perturbation will drive the system into a sequence of
incommensurate and commensurate phases and in a very tiny region near the RK point, the
commensurate phases occupy a set of infinitesimal measure. The sequence of incommensurate crystalline phases form something like a gapless, logarithmically confining “phase”.
This phenomenon, called Cantor deconfinement[55], is the default scenario unless the microscopic system is highly fine tuned (for example, to the RK QDM Hamiltonian). The
construction of the bipartite QDM phase diagram presented in the main text will realize
this Cantor deconfined scenario because of the small but nonzero error terms.
80
y
x
a
b
c
2
0
1
0
2
1
2
0
1
0
2
0
1
2
0
-1
0
1
-1
0
0
1
-1
0
-2
2
0
-2
-1
-3
-1
-3
-5
-7
-9
0
-4
-2
-1
-3
-6
-5
-8
-7
-9
Figure A.1: This figure gives three representative dimer coverings and the associated height
mapping. In each case, the first hexagon of the middle row was assigned height 0 and
the remaining values were obtained by the rules given in the text. In the columnar state
(a), the height is uniform in y for a given x and periodic in x with a period of 1 12 lattice
constants. The height field obtained by coarse graining will be uniform apart from this
microscopic fluctuation so is called a “flat state”. In contrast, the staggered state (c), is the
configuration having maximal tilt. (b) shows an arbitrary configuration.
81
2
2
1
0
1
0
2
2
1
0
0
0
-3
-1
-1
1
2
-2
-2
-3
-2
-1
0
3
0
3
0
-1
-4
-1
1
2
1
2
1
-2
-3
-2
0
3
0
3
0
-1
-4
-1
-3
Figure A.2: Periodicity of heights. The top figure shows a dimer pattern on the honeycomb
lattice where on the left and right side, the dimers are arranged in the same columnar
pattern. These regions are separated by a staggered region indicated by the circle. The
value of any local operator will depend only on the local dimer arrangement and thus
should be the same on the left and right of the circle. However, if we look at corresponding
plaquettes, we find that the values of the height variables on the right have been shifted
down by 3 relative to the left side. The precise value of the shift depends on the details
of the dimer pattern in the region separating the two columnar regions but the shift will
always be some integer multiple of 3. Therefore, the operator should be a periodic function
of the height variable with period 3. The bottom figure shows the similar construction for
the square lattice. In this case, the height construction involves gaining +3 if a dimer is
crossed when going clockwise around the sites of one of the sublattices and -1 otherwise.
Comparing corresponding plaquettes, it is seen that the periodicity is 4.
Appendix B
Mean field theory for string defects
B.1
Mean field calculation of the system response
Here we present some details of the mean field calculation outlined in section 2.4. In the
main text, it was shown that the system prefers the uniform density state when left to
itself. We now impose in each layer k a small external potential φkext (x), and study how the
system responds. We assume that this potential only acts on the positive charges, which are
the dynamical objects in our formulation. This requires adding a term to the Hamiltonian:
K
N
k=1
k
k
i=1 φext (xi )
and modifying the free energy functional (omitting terms that vanish
in a large N limit):
Fρ,N
T
=
K 1
k=−K
+
2
d2 xd2 x (ρk (x) − ρk+1 (x))(ρk (x ) − ρk+1 (x ))V (|x − x |)
d2 xρk (x) ln(
ρk (x)
)+
N
d2 xρk (x)φkext (x)
We minimize the functional subject to the normalization constraint,
(B.1)
2 k
d xρ (x) = N , to
obtain the self consistent equation:
N exp(−(φcext (x) + d2 x (2ρc (x ) − ρc+1 (x ) − ρc−1 (x ))V (|x − x |))
ρ (x) = 2
d x(exp(−(φcext (x) + d2 x (2ρc (x ) − ρc+1 (x ) − ρc−1 (x ))V (|x − x |))
c
(B.2)
We note that in the absence of a potential, we recover the uniform solution. If we start with
the uniform solution, then we may ask what happens when we turn on a small potential.
82
83
Specifically, we would like to calculate the susceptibility:
χx,x
c,c ≡ −
δρc (x) δφcext (x )
(B.3)
φcext (x)=0
The minus sign follows the convention of [32]. Differentiating Eq. B.2:
χx,x
c,c
=
N exp() δc,c δ(x − x )
d2 x exp()
δρc−1 (x ) δρc+1 (x )
δρc (x )
−
−
)V
(|x
−
x
|)
δφcext (x)
δφcext (x)
δφcext (x)
N exp()
− 2
d2 y exp()
( d x exp())2
δρc−1 (x ) δρc+1 (x )
δρc (x )
−
−
)
× δc,c δ(y − x ) + d2 x (2 c
δφext (x)
δφcext (x)
δφcext (x)
+
d2 x (2
× V (|y − x |)
φcext (x)=0
= ρ0 δ(x − x )δc,c −
−
(ρ0 )2 δc,c −
NT
,x
x ,x
d2 x (2χxc,c,x
− χxc−1,c
− χc+1,c )V (|x − x |)
,x
x ,x
d2 yd2 x (2χxc,c,x
− χxc−1,c
− χc+1,c )V (|y − x |)
1
x −x
= ρ0 (δ(x − x ) − )δc,c − d2 x (2χxc−c−x
− χxc−c−x
−1 − χc−c +1 )V (|x − x |)
A
1
x −x
x −x
d2 yd2 x (2χxc−c−x
−
χ
−
χ
)V
(|y
−
x
|)
(B.4)
+
c−c −1
c−c +1
A
where in the last line, we have assumed translational invariance. We also assumed that
the layers are sufficiently large (and that the susceptibilities are sufficiently “well-behaved”)
that the limits of integration may be taken to be the entire plane R2 . The following is seen
by integrating Eq. B.4:
d2 xχx−x
c−c = 0
(B.5)
This expresses the physical fact that, on average, the change in density due to the perturbing
potential is zero, which is another way of saying that charges can not leave the plane. Eq. B.5
allows a simplification of Eq. B.4:
x −x
d2 yd2 x (2χxc−c−x
− χxc−c−x
−1 − χc−c +1 )V (|y − x |) =
x −x −y
x −x −y
x −x −y
d2 yd2 x (2χc−c
− χc−c
− χc−c
−1
+1 )V (|x |) =
d2 x V (|x |)
x −x −y
x −x −y
x −x −y
d2 y(2χc−c
− χc−c
− χc−c
−1
+1 ) = 0
84
Therefore, we have an expression for the susceptibility (where we have chosen x = 0 and
c = 0) given by:
χxc = ρ0 (δ(x) −
1
)δc,0 −
A
d2 y(2χyc − χyc−1 − χyc+1 )V (|x − y|)
(B.6)
Suppose we apply a small external field in one of the layers. We would like to know how the
density responds in that plane and in the z-direction. Far away from the charge, the density
should be essentially the uniform equilibrium value i.e, ρk (x) = ρ0 . We are interested in
length scales associated with the decay of the perturbation in both the plane and z-direction.
We assume the perturbation induces a linear response in the density:
c
(δρ (x))[φext ] =
c
= −
d2 x
c
δρc (x) φcext (x )
c
φ
=0
δφext (x ) ext
c
d2 x χx−x
c−c φext (x )
Taking the Fourier transform gives:
δρ(q, qz ) = −χ(q, qz )φext (q, qz )
(B.7)
where the Fourier transform of Eq. B.6
χqqz
=
(1 − δq,0 )q 2
4πκ
1
q2
4πκρ0
+ 1 − cos qz
(B.8)
A potential acting in the c = 0 plane would have a functional form φcext (x) = f (|x|)δc,0 or
in momentum space,φext (q, qz ) = f (q). The corresponding density perturbation will be:
δρ(q, qz ) = −
where aq =
q2
4πκρ0
1
(1 − δq,0 )q 2 f (q)
4πκ
aq − cos qz
(B.9)
+ 1. We invert this in the z-direction:
π
dqz
δρ(q, qz )eiqz n
2π
(1 − δq,0 )q 2 f (q) π
eiqz n
= −
dq
z
4π 2 κ
2aq − eiqz − e−iqz
−π
δρ(q, n) =
−π
(B.10)
We make a variable change z = eiqz , dz = izdqz , and the integral becomes a contour integral
about the unit circle. (This is for n ≥ 0; for negative n, we need to use the variable change
85
z = e−iqz . Note that the expression is symmetric in n so we will get the same result either
way.)
δρ(q, n) =
where αq = aq +
(1 − δq,0 )q 2 f (q)
4π 2 κi
zn
(z − αq )(z − α−1
q )
dz
|z|=1
(B.11)
2
(aq − 1). Notice that if q = 0, aq > 1, αq > 1, and only the pole at
z = α−1
q lies inside the contour. Evaluating the residue, we find if n ≥ 0:
δρ(q, n) = −
= −
(1 − δq,0 )q 2 f (q) αq 1 2πκ
α2q − 1 αnq
(1 − δq,0 )q 2 f (q) αq
e−(ln αq )n
2πκ
α2q + 1
(B.12)
As suggested, we see that associated with each in-plane wavenumber q, we have an out of
plane length scale given by:
ξz (q) =
1
= ln αq
ln 1 +
1
q2
4πκρ0
+
q2
q2
4πκρ0 ( 4πκρ0
(B.13)
+ 2)
To say something about an in-plane length scale, we assume the perturbation is a point
charge (that acts only on the positive charges in the plane), i.e. f (q) =
2πκ
.
q2
We note that
close to our perturbing charge, the assumption of the perturbation being small will break
down. We can invert this, at least formally, to get the density perturbation in real space:
1
x
δρ( , n) = 2 2
ξ
4π ξ
B.2
is·( ξx )
e
2
d s
n−1
s2 (s2 + 2) 1 + s2 + 2 s2 (s2 + 2) 1 + s2 + s2 (s2 + 2)
(B.14)
Correlation lengths
As suggested, a natural in-plane length scale comes out of the calculation:
ξ =
1
1
)2
2Dκρ0
(B.15)
where κ = 1/2 and ρ0 is the density of defects in a layer. This scale depends on the
temperature and magnetic field through ρ0 . We may estimate ρ0 by considering the grand
partition function:
Z = e−βA =
y (2K+1)N
N
(N !)2K+1
eSN
(B.16)
86
where SN is the entropy of having N defects and y = e−Es /kB T is the fugacity of a positive
defect (y 2K+1 is the fugacity of a “string”) where Es = 8gµB JB/3 is the energy per kagometriangle bilayer of a string. At mean field level, we may replace SN by Sρ,N = (2K+1)N ln A.
Making this substitution, we obtain:
ZM F =
N
(yA)N 2K+1
N!
N
where N0 is the value for which
(yA)N
N!
≈
(yA)N0 2K+1
N0 !
(B.17)
is a maximum; the latter approximation holds if K
is large. Using Stirling’s approximation, we may estimate that N0 ≈ yA. We may then
estimate the average number of defects per layer:
N =
1
ZM F
N
N
(yA)N 2K+1
N!
≈ N0 ≈ yA
(B.18)
from which we conclude that:
2
ξ,M
F ∼ exp(8gµB JB/3kB T )
(B.19)
We can also say something about the out-of-plane correlation by looking at the density
perturbation (Eq. B.14) when x = ξ x̂. The result is:
1
δρ(x̂, n) = 2 2
4π ξ
eis·x̂
d2 s n−1
s2 (s2 + 2) 1 + s2 + 2 s2 (s2 + 2) 1 + s2 + s2 (s2 + 2)
(B.20)
The most striking thing about this result is that the rate of decay in the z-direction has no
dependence on temperature, although the coefficient of the integral does have a temperature
dependence. The implication is that when we place a charge at the origin; move away
by an in-plane correlation length, and then look upwards, the decay will be the same
regardless of temperature. Thus, there is no temperature-dependent length scale in the zdirection. This same conclusion comes out of the RG calculation, although with a different
temperature/field dependence.
Appendix C
Renormalization group treatment
of string defects
In this appendix, we provide details of the renormalization group calculation outlined in
section 2.4.
Let us begin by considering the canonical partition function Z({Nk,l }), corresponding
to the dipole distribution {Nk,l }. We would like to calculate, to order dτ , the result of
integrating out those modes where in some plane, an oppositely charged pair is separated
by a distance between τ and τ + dτ .
We introduce the abbreviation:
2 (1) 2 (2)
(1)
(2) d xk,i d xk,i xk,i − xk,i dΩτ =
τ2
k,i∈Ik
τ2
δ
τ
(C.1)
in terms of which the canonical partition function for a given dipolar distribution {Nk,l }
may be written: Z({Nk,l }, τ ) =
Ωτ
dΩτ exp(−H). Our RG calculation has two steps. The
first step is integrating over short length scales, i.e. those states where at least one pair
of charges is separated by a distance between τ and τ + dτ . The second step is to rescale
variables to restore the short distance cutoff. When we carry out the first step, the result
is a zeroth order term and a correction of order dτ :
Z({Nk,l }, τ ) =
Ωτ +dτ
dΩτ exp(−H) +
k,l,m,i,j
87
Iklmij
(C.2)
88
where Iklmij is the contribution of the configuration that has the negative end of the ith
m-dipole of layer k paired with the positive end of the jth (l − m)-dipole of layer k + m.
The sum over k is over all planes; the sum over l is over all dipole lengths up to the number
of planes; and the sum over m is from 1 to l − 1. The form of this term is given by:
Iklmij =
Ωτ +dτ
−H dΩτ e
A
(1)
(1)
(2)
(2)
(1)
(2) d2 xj xj − xj −H(x(2) ,x(1) )
d2 xi xi − xi i
j
δ
δ
e
(2)
τ2
τ
τ2
τ
d(xi ,τ )
(C.3)
(1)
The region of integration of the positive charge xj
is an annulus of radius τ and thickness
(2)
(2)
dτ centered on the negative charge xi . This region is denoted by d(xi , τ ). The position
of this negative charge (and hence the pair) is integrated over the entire area A. Strictly
(2)
speaking, xi
would have to avoid the hard cores of all of the other charges but this
introduces an error of order (dτ )2 . Ωτ +dτ is the space of configurations of the rest of the
charges in which the charges are separated from each other by a distance of at least τ + dτ .
(2)
(1)
(2)
H(xi , xj ) refers to the piece of the Hamiltonian which involves charges xi
(1)
and xj
and
the rest of the Hamiltonian is denoted by H .
(1)
(1)
(2)
(1)
The xj integration amounts to making the substitution xj = xi +
τ ; d2 xj
= τ dτ dθ;
and integrating over angles. If we denote the latter two of integrals of equation C.3 by I,
then:
I =
dτ
τ
A
(2)
d2 xi
τ2
2π
0
(2)
dθe−H(xi
(2)
,
xi +
τ)
δ
(2)
(2)
x(1) − x(2) xi − xj + τ i
i
τ
δ
τ
(C.4)
We assume that our gas of defects is sufficiently dilute that the following distances are
much greater than the pair separation τ : (1) the distance of a particle in plane k + m from
(1)
our pair, (2) the distance of a particle in plane k from the positive charge xi , and (3) the
(2)
distance of a particle in plane k + l from the negative charge xj . In this dilute limit, we
may make the approximation:
δ
(2)
(2)
(1)
(2)
xi − xi xi − xj + τ τ
δ
τ
(1)
(2)
τ 2 xi − xj ≈ δ
A
τ
(C.5)
89
(2)
(1)
We also have that H(xi , xj ) is small in this limit, which allows us to expand the exponential and to leading order, the integral may be done exactly [31]. The result is:
(1)
(2)
dτ xi − xj (πκτ 2 )2 rab δ
2π −
ea eb ln
τ
τ
A
τ
a=b
I =
(1)
(2)
− xj τ
dτ xi
≈ 2π δ
τ
(C.6)
In the penultimate line, the sum refers to a sum over all charges, positive and negative,
residing in the plane k + m. This sum term may be neglected in the large A limit, which
is why, in contrast to the Kosterlitz calculation[33], the coupling strength does not vary
during our RG flow (see equation 2.13). The delta function implies that the m-dipole and
(l − m)-dipole have been combined into a larger l-dipole. Returning to our correction term:
Iklmij
dτ
≈ 2π
τ
Ωk,l,m
τ +dτ
dΩk,l,m
exp(−H)
τ
(C.7)
where the space Ωk,l,m
τ +dτ is analogous to Ωτ +dτ , except that there is one less m-dipole in
layer k; one less (l − m)-dipole in layer k + m; and one more l-dipole in layer k. What we
are actually interested in is the grand partition function (equation 2.11). Because our RG
procedure is consistent with the charge neutrality constraint, the various {Iklmij } may be
combined with different terms in the grand partition function. When we substitute into
Eq. 2.11 and arrange terms, we find that:
Z =
{Nκ,λ }
×
1
κ,λ (Nκ,λ )!
yλ N,λ
κ,λ
2π
+
Ωτ +dτ
dΩτ exp(−H)
yλ N,λ k,l,m κ,λ
y Nk,l −1
dτ ym yl−m
l
2π
N
k,l
τ (2π)2
2π
2π
(C.8)
Nk,l −1
The prime on the second product means that yl
has been taken outside the product.
If the fugacities are small, then we may write this in a more convenient way:
Z =
N,l (yl + dτ l−1
m=1 ym yl−m )
τ
{Nk,l } k,l
(2π)Nk,l (Nk,l )!
Ωτ +dτ
dΩτ exp(−H)
(C.9)
90
Finally, we rescale lengths, x → x(1 + dτ /τ )−1 , and find (dropping primes):
Z=
{Nk,l } k,l
yl N,l 2π
(Nk,l )!
Ωτ
dΩτ exp(−H)
(C.10)
where
yl = (yl +
l−1
dτ dτ
dτ
ym yl−m )(1 + 2 )(1 − κ )
τ m=1
τ
τ
The flow equations (2.13) follow from this.
(C.11)
Appendix D
Sign conventions in the overlap
matrix
In the Rokhsar-Kivelson derivation of the dimer model, the overlap matrix between the
different dimer coverings plays a crucial role. Whether or not the RK point corresponds to
an equal-amplitude superposition of all dimer coverings depends on the question of whether
the dimer Hamiltonian can be turned into a form where all off-diagonal matrix elements
corresponding to the shortest resonance loops are negative.
The leading effect of a perturbing nearest-neighbour exchange finally gives rise to a
constant (for each type of loop) multiplying the overlap matrix (restricted to that type of
loop). If the overlap matrix can be written as a term proportional to matrix with entries
only 0 or 1, the ground state is indeed given by an equal-amplitude superposition at the
RK point.
In the following, we demonstrate that a fermionic sign convention may be used to generate the negative sign, independent of lattice details, for models involving valence bond
coverings of the lattice. We show that for the honeycomb and diamond lattices, we may
obtain the positive sign as well. We also present a convention for the Kivelson-Klein model
which gives the negative sign.
91
92
D.1
Overlaps in the fermionic convention
A general convention for overlaps can be obtained by employing the so-called fermionic
convention, where a valence bond between sites a and b is generated via operators, such as
the one placing a fermion with spin up on site a: c†a↑ . The singlet bond is then defined as:
1
|[ab] ≡ d†ab |0 = √ [c†a↑ c†b↓ + c†b↑ c†a↓ ]|0 .
2
(D.1)
Here |0 is the vacuum state with no fermions present. Note that
d†ab = d†ba ,
and that these operators, being bilinear in fermions, commute unless they have exactly one
site in common. This means that for constructing a valence-bond covering, the order in
which the bonds are generated is inconsequential.
A loop in the transition graph involving sites h, i, j, and k will lead to the following
type of expression in the overlap matrix element calculation:
0|dab · · · dij d†jk d†hi · · · d†ab |0
1
= − 0|dab · · · d†hk · · · d†ab |0
2
(D.2)
By induction, a loop in the transition graph involving L dimers in each configuration
leads to a factor of (−1/2)L−1 , independently of any further details of the lattice.
D.2
Honeycomb and diamond lattices
The following approach works for both honeycomb and diamond lattice, both of which have
a shortest resonance loop of length six, as in the original benzene picture.
As these lattices are bipartite, we can orient each bond to point from one sublattice (A)
to the other (B), so that a singlet between sites a and b of sublattices A and B, respectively,
has the wavefunction
√
|(ab) ≡ [| ↑a ↓b − | ↓a ↑b ] / 2 .
(D.3)
93
The two singlet coverings of the benzene loop now have wavefunctions
|1 ≡ |(ab)(cd)(ef ) ; |2 ≡ |(bc)(de)(f a)
from which it follows that
1|2 = +1/4
(D.4)
for any hexagonal plaquette.
It is in fact also possible to choose
1|2 = −1/4
(D.5)
for the honeycomb lattice. This can, for example, be achieved by choosing any fixed hardcore dimer covering of the triangular lattice which is dual to the honeycomb lattice under
consideration. One then multiplies each valence bond state of the honeycomb lattice by
(−1)n× , where n× is the number of valence bonds which cross dimers of the triangular
dimer covering. This generates the desired effect.
D.3
Other bipartite lattices
The above construction for generating uniform overlap matrix elements can be generalised
to any bipartite lattice. By orienting the bonds from one sublattice to the other, one always
obtains an overlap which is positive; its size is the simple product over the individual loops
involved in the transition graph.
k|l > 0 ;
(D.6)
indeed, the positive overlap holds true for any value of the loop length, and therefore for
an arbitrary pair of valence bond coverings |k, |l.
D.4
Kivelson-Klein model on pyrochlore lattice
Here we first need to establish the possible resonance loops. These involve six dimers on a
cluster of twelve sites arranged as follows.
94
Six of the sites sit on a hexagonal ring on the pyrochlore lattice; each link of this
hexagonal ring belongs to a different tetrahedron. Each of these six tetrahedra contains
one dimer linking a site on the hexagonal ring with a site off the hexagonal ring. As there
is a choice of 2 such off-sites per tetrahedron, the total number of shortest resonance loops
corresponding to each hexagonal ring equals 26 = 64. Not all of these loops are symmetry
equivalent.
These loops all involve moving six dimers. Hence, their overlap in the Fermion convention is given by −1/32.
Appendix E
Spinon gap for the decorated
honeycomb lattice
A stable RVB liquid phase requires not only certain properties of the ground state wavefunction but also that the excitation spectrum has a positive lower bound. In this section,
we argue that the nearest neighbor valence bond ground states are separated by a finite
gap from the excited states for the case of the decorated honeycomb lattice Klein model.
To this end observe that in the highly decorated limit we are connecting a set of
Majumdar-Ghosh[46] (MG) chains into a two-dimensional network. The excitations of
the chains themselves are well studied: these are spinon defects between the two different
dimerizations and there is considerable analytical[69, 70] and numerical[71] evidence that
they are gapped. In putting the chains together we need to ask if the intersections lead to
the emergence of states below the one-spinon continuum on the chains that can fill in the
gap. In the infinite decoration limit, it is sufficient to consider a single intersection: thus
we look for bound states localized near a site of the original honeycomb lattice where three
MG chains would cross (see Fig. E.1).
With reference to Fig. E.1, we consider the Klein Hamiltonian:
HK =
P̂N (i) .
i
95
(E.1)
96
4a
3a
2a
1a
0
2c
4c
1c
2b
1b
3b
3c
4b
Figure E.1: This figure depicts a spinon at the crossing of the edges of a decorated honeycomb lattice. An up spinon is at position 0 and the other spins are dimerized into singlet
pairs.
As before, i is the site index; N (i) is the “neighborhood” of i consisting of the site and
its nearest neighbors; and P̂N (i) projects the set of spins N (i) onto its highest spin state.
Because the highest spin state is symmetric under interchange of spins, if a wavefunction
involves spin i forming a singlet with one of its neighbors, this wavefunction will be destroyed
by the projector P̂N (i) . A projection operator has only non-negative eigenvalues so nearestneighbor valence bond states are ground states of the Hamiltonian (3.3). We consider a
wavefunction describing a single spin at the crossing of three semi-infinite MG chains.
|ψ0 = |+0 |00ee .
(E.2)
Here |+0 denotes an up spin at location 0 and |00ee denotes that “everything else” is
dimerized into nearest-neighbor singlet (00) pairs. We consider the action of the Klein
Hamiltonian (3.3) on this state. Observe that P̂N (0) , which involves only site 0 and its
three neighbors, is the one projector that does not destroy our trial function. We may write
this projector in terms of the spins involved:
P̂N (0) = Ccr (S 2 − 2)S 2 .
(E.3)
1a + S
1b + S
1c and Ccr is a positive constant which sets the energy scale for
=S
0 + S
Here S
97
violating the Klein condition at the crossing. Then:
S2 =
Si2 +
i · S
j
2S
i<j
i=0,1a,1b,1c
1 3
i · S
j
· ·4+
2S
2 2
i<j
=
= 3+
i · S
j .
2S
(E.4)
i<j
Therefore:
P̂N (0) = Ccr [3 +
i · S
j ][1 +
2S
i<j
= 3+4
i · S
j + (
2S
i<j
i · S
j ]
2S
i<j
i · S
j )(
2S
i<j
i · S
j )
2S
(E.5)
i<j
0 · S
1a (3 such terms),
The operator S 2 is made of operators involving the central site, 2S
1b (3 such terms). We
1a · S
and also interactions between neighbors of the central site, 2S
compute the actions of these terms on our trial state.
1a |ψ0 = |00ee 2S
0 · S
1a |+0 |001a,2a
0 · S
2S
=
=
=
=
=
1
−
+
z
√ |00ee [S0+ S1a
+ S0− S1a
+ 2S0z S1a
][0↑ (1a)↑ (2a)↓ − (1a)↓ (2a)↑ ]
2
1
1
1
√ |00ee [−0↓ (1a)↑ (2a)↑ + 0↑ (1a)↑ (2a)↓ + 0↑ (1a)↓ (2a)↑ ]
2
2
2
1 1
√ |00ee [ 0↑ (1a)↑ (2a)↓ − (1a)↓ (2a)↑ + 0↑ (1a)↓ (2a)↑ − 0↓ (1a)↑ (2a)↑ ]
2
2
1
|00ee [ |+0 |001a,2a + |+2a |000,1a ]
2
1
|ψ0 + |ψ2a
(E.6)
2
Here |ψ2a means a state having an up spin at site 2a with “everything else” dimerized into
singlet pairs.
1b |ψ0 = |00ee |+0 2S
1a · S
1b |001a,2a |001b,2b
1a · S
2S
=
1
+ S
z z
− − +
|00ee |+0 [S
1a 1b + S1a S1b + 2S1a S1b ][(1a)↑ (2a)↓ − (1a)↓ (2a)↑ ]
2
× [(1b)↑ (2b)↓ − (1b)↓ (2b)↑ ]
98
=
1
+ S
z z
− − +
|00ee |+0 [S
1a 1b + S1a S1b + 2S1a S1b ][(1a)↑ (1b)↑ (2a)↓ (2b)↓
2
− (1a)↑ (1b)↓ (2a)↓ (2b)↑
− (1a)↓ (1b)↑ (2a)↑ (2b)↓ + (1a)↓ (1b)↓ (2a)↑ (2b)↑ ]
=
+
+
=
1
|00ee |+0 [−(1a)↑ (1b)↓ (2a)↑ (2b)↓ − (1a)↓ (1b)↑ (2a)↓ (2b)↑
2
1
1
(1a)↑ (1b)↑ (2a)↓ (2b)↓ + (1a)↑ (1b)↓ (2a)↓ (2b)↑
2
2
1
1
(1a)↓ (1b)↑ (2a)↑ (2b)↓ + (1a)↓ (1b)↓ (2a)↑ (2b)↑ ]
2
2
1
1
|00ee |+0 [ (1a)↑ (2a)↓ + (1a)↓ (2a)↑ (1b)↑ (2b)↓ + (1b)↓ (2b)↑
2
2
− (1a)↑ (2a)↑ (1b)↓ (2b)↓ − (1a)↓ (2a)↓ (1b)↑ (2b)↑ ]
=
1
|00ee |+0 |101a,2a |101b,2b − |111a,2a |1, −11b,2b
2
− |1, −11a,2a |111b,2b
(E.7)
In the previous line, the notation |10i,j means that spins i and j are paired in their m = 0
triplet state and so forth. The action of S 2 on |ψ0 generates the two types of terms described
above. Note that (E.7) is orthogonal to |ψ0 . The effect of acting with S 2 a second time
may be found by examining its effect on the terms in equations (E.6) and (E.7). Many of
the generated terms will be orthogonal to |ψ0 but some will not be including, for example:
1a |ψ2a = |+2a |00ee 2S
0 · S
1a [|000,1a ]
0 · S
2S
3
= − |ψ2a
2
(E.8)
and
1b
1a · S
2S
1
1
+ S
z z
− − +
|00ee |+0 |101a,2a |101b,2b = |00ee |+0 [S
1a 1b + S1a S1b + 2S1a S1b ]
2
4
(1a)↑ (2a)↓ + (1a)↓ (2a)↑
=
+
=
=
(1b)↑ (2b)↓ + (1b)↓ (2b)↑
1
|00ee |+0 (1a)↑ (2a)↑ (1b)↓ (2b)↓ + (1a)↓ (2a)↓ (1b)↑ (2b)↑
4
1
(1a)↑ (2a)↓ − (1a)↓ (2a)↑ (1b)↑ (2b)↓ − (1b)↓ (2b)↑
2
1
|00ee |+0 |001a,2a |001b,2b + |111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b
4
1
1
|ψ0 + |00ee |+0 |111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b
(E.9)
4
4
99
and similarly,
1b
1a · S
2S
=
−
=
−
=
1
|00ee |+0 (|111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b )
2
1
|00ee |+0 (1a)↑ (2a)↓ (1b)↓ (2b)↑ + (1a)↓ (2a)↑ (1b)↑ (2b)↓
2
1
(|111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b )
2
1
|00ee |+0 |101a,2a |101b,2b − |001a,2a |001b,2b
2
1
(|111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b )
2
1
1
− |ψ0 + |00ee |+0 2|101a,2a |101b,2b − |111a,2a |1, −11b,2b
2
4
− |1, −11a,2a |111b,2b
(E.10)
Therefore, we find that:
P̂N (0) |ψ0 = Ccr
54
4
|ψ0 + 4 |ψ2a + |ψ2b + |ψ2c + |α
(E.11)
where |ψi = |+i |00ee describes an up spin at location i with everything else paired into
singlets; Ccr sets the energy scale for violating the Klein condition at the cross; and |α is a
piece orthogonal to the spinon states, {|ψi }. We may also compute the effect of operator
(3.3) on the state |ψi , where i is a location along a chain; in this case, P̂N (i) is the only
operator that does not destroy the state:
P̂N (i) |ψi = Cch
5
2
|ψi + |ψi−2 + |ψi+2 + |β
(E.12)
Cch is the energy scale for violating the Klein condition on the chain and |β is a piece
orthogonal to the spinon states. We see that the Klein Hamiltonian acting on a spinon
state produces the original state; spinon states where the spin has hopped two sites; and
terms orthogonal to all spinon states. This motivates the bound state trial function:
|ψ = |ψ0 +
y n |ψ2n,i
(E.13)
n>0;i=a,b,c
where y is a variational parameter less than unity. We are interested in the expectation
value of the energy for this state (E.13):
E=
ψ|HK |ψ
ψ|ψ
(E.14)
100
In calculating this, we need to contend with the non-orthogonality of the spinon states:
i ψ|ψj
1 |i±j|/2
= −
(E.15)
2
where the minus (plus) is for sites i and j on the same (different) chain(s). First, we compute
the normalization:
ψ|ψ = 1 + 2
n>0;i=a,b,c
= 1 + (2 · 3)
y
1 n
n
n>0
= 1+6
ψ|HK |ψ
Ccr
−
2
y n+m (2m,i ψ|ψ2n,j )
n,m>0;i,j=a,b,c
+6
1 m+n
y n+m −
m,n>0
2
+3
1 |m−n|
y n+m −
m,n>0
−y/2 2
−y/2
3y 2 (1 − y/2)
+6
+
1 + y/2
1 + y/2
(1 + y/2)(1 − y 2 )
54
= ψ|
4
|ψ0 + 4 |ψ2a + |ψ2b + |ψ2c
+ A
yn
n>0,i=a,b,c
54
= ψ| (
+ A
=
y n (0 ψ|ψ2n,i ) +
4
5
2
2
(E.16)
|ψ2n,i + |ψ2n−2,i + |ψ2n+2,i
+ 3Ay)|ψ0 + (4 − A) |ψ2a + |ψ2b + |ψ2c
5
( y n + y n−1 + y n+1 )|ψ2n,i
2
n>0,i=a,b,c
5
y
3
1
54
54
+ 3Ay + (4 − A)(− ) + A[ + + y] +
+ 3Ay · 3
(− )n
4
2
2 y
4
2
n>0
+ (4 − A)
1
1
y n [(− )n−1 · 3 + (− )n+1 · 6]
2
2
n>0
5 1
y n+m (2m,i ψ|ψ2n,j )
+ A( + + y)
2 y
n,m>0;i,j=a,b,c
=
+
+
=
+
54 − 6(4 − A)
+ 3Ay
4
54 + 10A
−y/2 A
+ + 4Ay − 6(4 − A) − 3(4 − A)
3
4
y
1 + y/2
5
2
1
3y (1 − y/2)
−y/2 2
A + + y [6
+
2 y
1 + y/2
(1 − y 2 )(1 + y/2)
18 + 66A
−y/2 3A
30 + 6A
+ 3Ay +
+
+ 12Ay
4
4
y
1 + y/2
5A
2
A
3y (1 − y/2) −y/2 2
+ + Ay 6
+
2
y
1 + y/2
(1 − y 2 )(1 + y/2)
(E.17)
101
where A = Cch /Ccr . Fig. E.2 shows a graph of the expectation value of the energy for
several values of A. For small values of A, the best variational wavefunction is an extended
scattering state while for large values, the best wavefunction is localized at the cross.
30
a=0.5
a=1
a=2
a=5
a=20
25
E
20
15
10
5
0
0
0.2
0.4
0.6
0.8
1
y
Figure E.2: This plot gives the energy expectation as a function of the variational parameter
y for different values of A = Cch /Ccr , which is ratio of Klein scales; the energy is measured in
units of Ccr . For large values of this ratio, the minimum occurs for y = 0 which corresponds
to spinons localized at the crossing. For small values, the minimum occurs at y = 1, which
corresponds to an extended spinon state. At a value slightly greater than A = 1, there is a
first order “phase transition” between these limits. The important feature is that for any
nonzero A, there is an energy gap between spinon states and the valence bond states, which
have zero energy.
Our analysis has been for the infinite decoration limit. For large, but finite, decoration,
the extended spinon states obtained for small A may be interpreted as MG spinons scattering
at the vertices. In this limit, the natural extension of the localized states obtained for large
A will involve the wavefunction having peaks at the reference sites and decaying on the
chains. There will be a band of such localized states below the scattering states. We may
estimate the bandwidth by considering a variational wavefunction where the spinon resides
102
only on the reference sites:
|Ψ =
eik·n |
n
(E.18)
n
where |
n denotes a wavefunction of the form (E.2) for the spinon at lattice site n and k is
a wavenumber. The variational energy of this trial state may be calculated to leading order
in a large N expansion, where N is the number of sites inserted between reference sites in
the decoration.
Ψ|HK |Ψ
Ccr Ψ|Ψ
=
=
≈
&×
Ψ|
54
na + Ψ|
nb + Ψ|
nc +4
eik·n
4
Ψ|Ψ
n
m|
54
na + m|
nb + m|
nc +4
eik·(n−m)
4
Ψ|Ψ
n,m
√
√
kx
54
kx
3ky
3ky 2N
+ 16 cos kx + cos( +
) + cos( −
) (x + 2x2N +2 )
4
2
2
2
2
n
Ψ|Ψ
≈
54
kx
+ 16x2N cos kx + cos( +
4
2
√
kx
3ky
) + cos( −
2
2
√
3ky )
2
Here |
na is a state with an up spinon on the a-chain at site n, etc. and x =
(E.19)
√1 .
2
We see
that the band becomes more narrow as the decoration is increased.
In our analysis so far, we have considered states where the defects are always an even
number of sites away from the reference sites. There is another family of spinon states
corresponding to the defects being located on the odd sites. Referring to Fig. E.3, the
Klein operator permits the spin at 1a to hop only to the site 1b, which is connected by a
dimer to the origin. Therefore, in the large decoration limit, this is equivalent to the MG
chain, which we know is gapped.
103
4a
3a
2a
1a
0
2c
4c
3c
1c
2b
1b
3b
4b
Figure E.3: This shows a representative of the family of states where a spinon is an odd
distance from the origin. In this configuration, the Klein operator may hop the spinon only
onto the b-chain. The configuration where the origin forms a singlet with 1c is essentially
orthogonal to the given configuration for large decoration.
Appendix F
Classical dimers on the pentagonal
lattice
At infinite temperatures, where “infinite” means a temperature that is small compared
with the excitation gap of the Klein Hamiltonian but much larger than the energy scales of
the quantum dimer Hamiltonian, the dimers are described classically, i.e. thermodynamic
quantities are computed as equal-weight averages over all dimer states. The number of
dimer states grows exponentially with lattice size. This number may be computed using
the method of Kasteleyn [79, 80]. The results are shown in Table F.1. The method also
yields the entropy per site in the thermodynamic limit.
S = 0.168608 . . .
(F.1)
The striking feature of table F.1 is that even fairly small systems have an enormous
number of dimer coverings so numerical studies of large systems require Monte-Carlo simulations. Fig. F.1 is a Monte-Carlo calculation of monomer-monomer correlation functions
C(r) for the pentagonal lattice using the algorithm of Sandvik[81]. C(r) is defined as the
number nd (r) of dimer coverings given a pair of test monomers separated by distance r
divided by nd (1). The simulation shows monomer deconfinement at infinite temperatures,
as opposed to the square and honeycomb lattices which show logarithmic confinement. This
104
105
1.1
along (1,0) direction
along (1,1) direction
C(r)
1.05
1
0.95
0.9
1
10
r
Figure F.1: The monomer-monomer correlation function for a 100x100 pentagonal lattice
with periodic boundary conditions. The two curves are cuts along the x̂ and x̂+ ŷ directions.
The distance r refers to the distance between unit cells. In computing these correlation
functions, we take the two test monomers to be on the same sublattice (the pentagonal
lattice is a cubic lattice with a six point basis). Each data point is an average over N√= 106
data points and the noise seen in the plot is of the order of Monte Carlo noise 1/ N ∼
10−3 ∼ 0.1%.
indicates a liquid phase at high temperatures. The RK point also has deconfined monomers
and the same dimer correlations as the high temperature phase, which strongly suggests
that the RK point is part of a zero temperature liquid phase which connects continuously
to the high temperature liquid.
106
Lx
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
Ly
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
nd
4
12
28
68
164
396
12
136
1068
9488
86252
798856
28
1068
17836
373412
7732928
160648524
68
9488
373412
21643648
1235195428
70937630864
164
86252
7732928
1235195428
192674444864
30315148743302
396
798856
160648524
70937630864
30315148743302
Table F.1: Table of the number of dimer configurations for an Lx xLy size pentagonal
lattice. Here Lx and Ly refer to the underlying square lattice; the pentagonal lattice is a
square lattice with a 6 point basis. We have assumed periodic boundary conditions in this
calculation.
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