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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
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c Copyright 2005 by Kumar S. Raman.All rights reserved.
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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 three-
dimensionality.
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 mod-
els 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
Roderichs careful analysis of
technical details which range from improving the noise of Monte
Carlo simulations to un-
derstanding the workings of an RG calculation. The work on spin
ice was done in collabora-
tion with Dr. Sergei Isakov (Toronto). While this thesis
emphasizes my contribution to that
effort, some of Sergeis 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 exten-
sion 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 Gre-
gor, 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 fathers side and my
grandfather, P. V. Chandra,
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on my mothers 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 . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 6
2.2 Model and notation . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 11
2.3 The two [111] magnetization plateaus . . . . . . . . . . . .
. . . . . . . . . 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
2.4 The low field regime . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 182.4.1 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
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2.7.1 Cooling by adiabatic (de)magnetization . . . . . . . . . .
. . . . . . 28
2.8 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
3.4 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
3.5 Dynamical selection of gauge structures: pyrochlore lattice
. . . . . . . . . . 57
3.5.1 The Klein model . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 57
3.5.2 The Kivelson-Klein model . . . . . . . . . . . . . . . . .
. . . . . . . 58
3.6 Discussion and outlook . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 60
A An overview of height representation theory 70
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 . . . . . . . . . . . . . . . .
. . . . . . . . 78
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
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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 consid-
ered 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 degen-
eracy 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 interac-
tion 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
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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. Onthe square lattice, the
interaction is optimally satisfied by the Neel configuration
drawnabove. In contrast, spins on the triangular lattice are
frustrated in that the third spinis unable to simultaneously
satisfy its interaction with its up and down neighbors.
Thisscenario does not arise on the square lattice because in that
geometry two neighboringspins 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 exam-
ple 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 in-
dependent 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.
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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 representa-
tion 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 thatwhile 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
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4
local resonance moves.
Spin liquids are characterized by rapidly decaying correlations,
translational and rota-
tional 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 ex-
ample, 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 respectivechapters 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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5
C1
C1
Figure 1.2: A valence bond state is a state where each spin
forms a singlet pair with one ofits neighbors. The RVB spin liquid
is a quantum superposition of all valence bond statesconnected by
local resonance moves. The above example shows a valence bond
covering fora part of a triangular lattice and the dotted lines
depict the simplest local move. C1 is aline extending through the
system and we see that initially there are three bonds crossingthe
line and after the flip, there is one. By inspection, we see that
if the number of bondscrossing the line is odd (or even) then this
property will not be affected by local resonancemoves. 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/verticalline may be odd/odd, odd/even,
even/odd, or odd/odd). The RVB spin liquid state is asuperposition
of all valence bond coverings in a given topological sector so the
state maybe labelled by its winding number. This global property is
called topological order.
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Chapter 2
The magnetization process of spinice 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 Ho2Ti2O7 and Dy2Ti2O7. 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 b ecomes 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 configu-
ration 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.
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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] directionsof the
pyrochlore lattice are the body diagonals of the cube, denoted by
d, and indicatedby 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 twoOs. 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
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experiments.
Pauling proposed that while the true ground state of ice may be
unique, thus satisfy-
ing 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 Paulings proposal may have
a magnetic analogy
and theoretically predicted spin ice roughly forty years before
its first experimental real-
ization. The discovery[9] was made when it was observed that the
pyrochlore compound
Dy2Ti2O7 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 Paulings prediction for water ice.
Spin ice is easier to work with (experimentally) than water ice
and may be used to ex-plore 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 unicrys-
talline 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 magneti-
zation fraction of 0.5 is the highest value consistent with the
easy axis constraint. The lower
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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 kindsof 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.
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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 =(i,j)
JijSi Sj + D(i,j)
3(Si rij)(Sj rij) Si Sjr5ij
Ei
d(i) Si2 +
i
B Si, (2.1)
The sums on (i, j) are over all pairs of spins; the spins {Si}
are unit-length. The first termis an exchange interaction where the
constants Jij are expected to decay
