Classical Spin Liquids and Floquet-Anderson Insulators

149 Pages·2017·4.22 MB·English

Frustration and Disorder:
Classical Spin Liquids and
Floquet-Anderson Insulators
Dillon T. Liu
Somerville College
University of Oxford
A thesis submitted for the degree of
Doctor of Philosophy
Trinity Term 2017
Abstract
In this thesis, the physics of two systems is investigated, both of which
support unconventional phases of matter. We investigate a frustrated
system which has a classical spin liquid regime in the paramagnetic phase.
Wealsoconsideraperiodically-driven, disorderedsystemandexplorenon-
equilibrium regimes. Both systems exhibit macroscopic behaviour which
differs strongly from the behaviour in standard scenarios.
First, the properties of frustrated magnets and related systems are dis-
cussed. We introduce three-dimensional generalisations of a well-studied
two-dimensional frustrated system. These generalisations consist of cou-
pled layers of two-dimensional triangular lattice Ising antiferromagnets.
We study three stackings that have nearest-neighbour interactions: two
frustrated stackings (abc and abab) and the unfrustrated stacking (aaa).
We use a combination of methods, including numerics and analytics, to
show that these generalisations exhibit a strongly correlated, highly fluc-
tuating regime known as a classical spin liquid. We show this by investi-
gating the structure factor and correlations directly and by describing a
mapping to a continuum field theory that gives further clarity about the
classical spin liquid regime using a renormalisation group analysis and
consideration of low-lying excitations in these systems.
Second, we study the response of a disordered system to a time-periodic
driving potential. We start in the context of Mott’s work on a.c. conduc-
tivity in Anderson insulators and move beyond this to study the Floquet
(long-time) regimes beyond linear response. Using Landau-Zener physics,
we construct a description of the various regimes which are possible. We
also present the results of thorough numerical studies of these systems.
Acknowledgements
I am thankful for everything that my supervisor, John Chalker, has
done for me over the past four years. Of course, this includes immense
contributions to the research I present in this thesis and all of my work
at Oxford. His insights and perspectives on physics and problem solving
have set a high bar that I aspire to meet.
I am also grateful for the camaraderie of my fellow students. Thanks
to Thomas Veness, Richard Fern, Stefan Groha, Bruno Bertini, Ga´bor
Hala´sz, Fenner Harper, Curt von Keyserlingk, Neil Robinson, Thomas
Scaffidi, Amos Chan, Jack Kemp, and Abi Kulshreshtha.
Thanks to other department members, including Adam Nahum, Arijeet
Pal, Zohar Ringel, Stefanie Thiem, and especially Dima Kovrizhin for
reading a draft of this thesis. Fabian Essler, Paul Fendley, and Steve
Simon are due thanks for wise advice. I am also thankful for mentors I
had as an undergraduate: Rahul Roy, Tom Intrator, Brian Greene, Max
Lipyanskiy, and John DiTusa.
Of course, I deeply appreciate the Marshall Aid Commemoration Com-
mission, Somerville College, and the Department of Theoretical Physics
for funding me and the Kavli Institute for Theoretical Physics for hosting
me during Autumn 2015.
I am grateful for support I received outside of physics as well. Thanks
to SCBC for putting me in a boat and to OULRC for showing me how
to move one. Contending with an ocean of separation from friends and
family in the US was not always easy, but visits and trips made that
bearable. I am also very thankful for the friendships I have forged while
in the UK. Finally, I will always owe my siblings and parents for their love
and support.
Author Contributions
The work presented in this thesis is based on a publication [1] and a
manuscript currently in preparation [2].
The author of this thesis contributed critically to [1] via the extensive
Monte Carlo simulations and the development of the self-consistent Gaus-
sian approximation treatment in that work. This included writing the
full parallel tempering Metropolis Monte Carlo simulations and carrying
out all of the data analysis for the results of those simulations. Addition-
ally, some independent numerical simulations were carried out by L. D.
C. Jaubert and used to corroborate results in Chapter 5.
[1] contains work which was principally contributed by F. J. Burnell
and J. T. Chalker (see also [3]), but which is included in this thesis in
Chapters 6, 7. These results are included in detail in this thesis because
they provide essential context for the problem on which Part II focuses.
Part III also contains work from a collaboration with V. Khemani, S.
L. Sondhi, and J. T. Chalker [2]. The author of this thesis carried out
essentially all the numerical work in this project and made significant con-
tributionstoidentifyingtheappropriatephysicalpropertiesofthesystems
and their signatures in the numerical simulations.
Contents
I Introduction 1
1 Overview and Outline 2
II Frustration 4
2 Frustrated Magnetism 5
2.1 Types of Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Properties of Frustrated Systems . . . . . . . . . . . . . . . . . . . . 7
2.3 Outline of Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Stacked Triangular Lattice Ising Antiferromagnets 13
3.1 Experiments and Previous Work . . . . . . . . . . . . . . . . . . . . . 15
4 Model and Mean-field Results 20
4.1 Mean-field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.1 Mean-field Predictions . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Self-consistent Gaussian Approximation . . . . . . . . . . . . . . . . . 25
5 Numerical Results 28
5.1 Monte Carlo Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.1 Parallel Tempering . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Implementation and Equilibration . . . . . . . . . . . . . . . . . . . . 33
5.3 Phase Diagram and Ordering Transition . . . . . . . . . . . . . . . . 36
5.4 Structure Factor and Correlations . . . . . . . . . . . . . . . . . . . . 39
5.4.1 abc stacking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4.2 abab stacking . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4.3 Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Comparison with self-consistent Gaussian approximation . . . . . . . 50
6 Height Models 52
6.1 Triangular Lattice Ising Antiferromagnet (TLIAFM) . . . . . . . . . 53
6.2 Coupled Height Models . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.3 Spin Correlations from Heights . . . . . . . . . . . . . . . . . . . . . 68
i
7 Renormalisation Group Analysis 71
7.1 Renormalisation Group for TLIAFM . . . . . . . . . . . . . . . . . . 71
7.2 Weakly Coupled Layers . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.3 Harmonic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.4 Bound Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.5 Transition to Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8 Phase Diagram and Summary 84
8.1 Phase Boundaries and Comparisons . . . . . . . . . . . . . . . . . . . 84
8.2 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 86
III Disorder 88
9 Disordered and Driven Systems 89
9.1 Anderson Localisation . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.1.1 Scaling Argument for Localisation . . . . . . . . . . . . . . . . 91
9.1.2 Transfer Matrices and Localisation . . . . . . . . . . . . . . . 93
9.1.3 Localisation and Numerics . . . . . . . . . . . . . . . . . . . . 94
9.2 Driving and Response in Quantum Systems . . . . . . . . . . . . . . 95
9.3 Outline of Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10 Floquet-Anderson Insulators 100
10.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.2 Models and Numerical Simulation . . . . . . . . . . . . . . . . . . . . 102
11 Weak Driving 108
11.1 The Golden Rule and Mott Conductivity . . . . . . . . . . . . . . . . 109
11.1.1 Single-site Drive . . . . . . . . . . . . . . . . . . . . . . . . . . 111
11.2 Long-time Limit from Rabi Problem . . . . . . . . . . . . . . . . . . 112
11.3 Numerical Results for Weak Driving . . . . . . . . . . . . . . . . . . . 113
12 Strong Driving and Long Times 120
12.1 Length Scales and Adiabaticity . . . . . . . . . . . . . . . . . . . . . 121
12.1.1 Adiabaticity and Landau-Zener Crossings . . . . . . . . . . . . 123
12.2 Strong Driving Regimes . . . . . . . . . . . . . . . . . . . . . . . . . 124
12.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
12.3.1 Breakdown of Linear Response . . . . . . . . . . . . . . . . . 128
12.3.2 Sample-to-sample and Site-to-site Fluctuations . . . . . . . . . 130
12.3.3 System Activity . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12.3.4 Adiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
12.4 Outlook and Further Questions . . . . . . . . . . . . . . . . . . . . . 136
13 Summary 138
ii
Part I
Introduction
1
Chapter 1
Overview and Outline
The figurative condensed matter physics diaspora, rooted in the study of solids and
liquids,isnowwidelyspreadacrossfarmoreexoticmatter. Thecoreprincipleremains
understanding how macroscopic properties emerge from simple models. In this thesis,
we use statistical physics, electromagnetism, and quantum mechanics to study two
examples of exotic matter. The presence of microscopic wrinkles in our simple models
givesrisetothedistinctivepropertiesofthesesystems. Thewrinklesare, respectively,
frustration and disorder. While the colloquial meanings of these words belie the
deeper nuances, their usage is in good faith.
In condensed matter physics, a primary focus is developing a framework to dis-
tinguish different kinds of matter, or phases. Huge amounts of effort have been spent
trying to understand, in a precise way, the manner in which types of matter differ
from each other. Symmetry and topology play key roles here and, over the past
century, have provided an impressively thorough framework for classifying phases of
matter in equilibrium. However, we will see that it is also fruitful to think beyond the
usual paradigm and investigate regimes within definite phases of matter or regimes in
non-equilibrium situations. In the case of the former, we will see that it is possible to
have dramatically different behaviour within a single phase of matter and in the case
of the latter, we will see that some systems have distinctive non-equilibrium regimes
and crossovers between these regimes.
In the frustrated and disordered systems we study, the essential physics of these
regimes is difficult to capture by considering only individual particles. Instead, one
must appeal to other methods. Many conceptual and practical developments have
2
beencrucialtoanysuccesswehavehadindescribingfrustratedordisorderedsystems.
Chief among these are two different approaches that provide insight to increasingly
large systems. One of these is the idea of scale invariance and the other is the con-
tinued increase in computational power available. The latter is an example of how
condensed matter physics is the gift that keeps on giving: solid state and semicon-
ductor physics gave rise to the transistor in the first half of the 20th century and the
technological advances that followed have opened vast avenues of research into ever
more sophisticated systems. The idea of scaling was developed more recently and
provides a theoretical apparatus to describe physics of condensed matter systems at
increasing length scales.
Althoughtheworkhereistheoretical, theimportanceofexperimentsincondensed
matter physics cannot be understated. A blessing of condensed matter physics is
the robust and frequent feedback between theoreticians and experimentalists. While
highlysophisticated,themanageablescaleofexperimentsincondensedmatterphysics
means that analytic and numerical progress is mostly anchored in reality.
In Part II, we present a study of three-dimensional geometrically frustrated mag-
nets. We give a brief, broad overview of frustrated magnetism in Chapter 2 to in-
troduce the topic. In the rest of Part II, we specifically study systems of stacked
triangular lattice Ising antiferromagnets and unconventional phases of matter that
arise therein.
In Part III, we study driven Anderson insulators, or Floquet-Anderson insulators.
We begin in Chapter 9 by giving some background about the kind of disordered
systems we study and introduce other tools used for studying periodically driven
quantum systems. Within the remainder of Part III, we study Anderson insulators
in the strongly-localised regime and with periodic, monochromatic drives and the
dynamical regimes that develop in these systems.
3
Part II
Frustration
4