From 0393ef0f2ff698a1e74c37a5bcc0b7b8c47c2049 Mon Sep 17 00:00:00 2001 From: Tom Hodson Date: Wed, 24 Aug 2022 19:03:28 +0200 Subject: [PATCH] updates --- _thesis/1_Introduction/1_Intro.html | 222 +++++++++++++----- .../4.2_AMK_Methods.html | 65 +++++ 2 files changed, 234 insertions(+), 53 deletions(-) diff --git a/_thesis/1_Introduction/1_Intro.html b/_thesis/1_Introduction/1_Intro.html index 1b3987e..db6afba 100644 --- a/_thesis/1_Introduction/1_Intro.html +++ b/_thesis/1_Introduction/1_Intro.html @@ -574,61 +574,72 @@ spin couples. In certain transition metal based compounds, such as those based on Iridium and Rutheniun, crystal field effects, strong spin-orbit coupling and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with -strongly anisotropic spin-spin couplings  [42].

-

The celebrated Kitaev model  [42,46–49]. Kitaev +materials draw their name from the celebrated Kitaev Honeycomb Model as +it is believed they will realise the QSL state via the mechanisms of the +Kitaev Model.

+

The Kitaev Honeycomb model  [46]

-

QSLs are a long range entangled ground state of a highly -frustated

- -

Geometric frustration or spin-orbit coupling can prevent magnetic -ordering is an important part of getting a QSL, suggests exploring the -lattice and avenue of interest.

- -

kinds of mott insulators: Mott-Heisenberg (AFM order below Néel -temperature) Mott-Hubbard (no long-range order of local magnetic -moments) Mott-Anderson (disorder + correlations) Wigner Crystal

+role="doc-biblioref">50] was the first concrete model with a +QSL ground state. It is defined on the honeycomb lattice and provides an +exactly solvable model whose ground state is a QSL characterized by a +static \(\mathbb Z_2\) gauge field and +Majorana fermion excitations. It can be reduced to a free fermion +problem via a mapping to Majorana fermions which yields an extensive +number of static \(\mathbb Z_2\) fluxes +tied to an emergent gauge field. The model is remarkable not only for +its QSL ground state, it supports a rich phase diagram hosting gapless, +Abelian and non-Abelian phases and a finite temperature phase transition +to a thermal metal state  [51]. It has also been proposed that it +could be used to support topological quantum computing  [52].

+

It is by now understood that the Kitaev model on any tri-coordinated +\(z=3\) graph has conserved plaquette +operators and local symmetries  [53,54] which allow a mapping onto effective +free Majorana fermion problems in a background of static \(\mathbb Z_2\) fluxes  [55–58]. +However, depending on lattice symmetries, finding the ground state flux +sector and understanding the QSL properties can still be +challenging  [59,60].

+

paragraph about amorphous lattices

+

In Chapter 4 I will introduce a soluble chiral amorphous quantum spin +liquid by extending the Kitaev honeycomb model to random lattices with +fixed coordination number three. The model retains its exact solubility +but the presence of plaquettes with an odd number of sides leads to a +spontaneous breaking of time reversal symmetry. I unearth a rich phase +diagram displaying Abelian as well as a non-Abelian quantum spin liquid +phases with a remarkably simple ground state flux pattern. Furthermore, +I show that the system undergoes a finite-temperature phase transition +to a conducting thermal metal state and discuss possible experimental +realisations.

Outline

-

This thesis is composed of two main studies of separate but related -physical models, The Falikov-Kimball Model and the Kitaev-Honeycomb -Model. In this chapter I will discuss the overarching motivations for -looking at these two physical models. I will then review the literature -and methods that are common to both models.

-

In Chapter 2 I will look at the Falikov-Kimball model. I will review -what it is and why we would want to study it. I’ll survey what is -already known about it and identify the gap in the research that we aim -to fill, namely the model’s behaviour in one dimension. I’ll then -introduce the modified model that we came up with to close this gap. I -will present our results on the thermodynamic phase diagram and -localisation properties of the model

-

In Chapter 3 I’ll study the Kitaev Honeycomb Model, following the -same structure as Chapter 2 I will motivate the study, survey the -literature and identify a gap. I’ll introduce our Amorphous Kitaev Model -designed to fill this gap and present the results.

-

Finally in chapter 4 I will summarise the results and discuss what -implications they have for our understanding interacting many-body -quantum systems.

+

The next chapter, Chapter 2, will introduce some necessary background +to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and +localisation.

+

In Chapter 3 I introduce the Long Range Falikov-Kimball Model in +greater detail. I will present results that. Chapter 4 focusses on the +Amorphous Kitaev Model.

@@ -984,13 +995,118 @@ class="csl-right-inline">H.-H. Lin, L. Balents, and M. P. A. Fisher, Symmetry in the Weakly-Interacting Two-Leg Ladder, Phys. Rev. B 58, 1794 (1998).
+
+
[46]
G. +Jackeli and G. Khaliullin, Mott Insulators in +the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum +Compass and Kitaev Models, Physical Review Letters +102, 017205 (2009).
+
+
+
[47]
M. +Hermanns, I. Kimchi, and J. Knolle, Physics +of the Kitaev Model: Fractionalization, Dynamic Correlations, and +Material Connections, Annual Review of Condensed Matter Physics +9, 17 (2018).
+
+
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[48]
S. +M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. +Gegenwart, and R. Valentí, Models and Materials for Generalized +Kitaev Magnetism, Journal of Physics: Condensed Matter +29, 493002 (2017).
+
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[49]
H. +Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, +Concept and Realization of Kitaev Quantum Spin Liquids, Nature +Reviews Physics 1, 264 (2019).
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[50]
A. Kitaev, Anyons in an Exactly Solved Model and Beyond, Annals of Physics 321, 2 (2006).
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[51]
C. +N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, Thermally Induced +Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of +the Kitaev Model with Parity Projection, Phys. Rev. B +99, 045142 (2019).
+
+
+
[52]
M. +Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological Quantum +Computation, Bull. Amer. Math. Soc. 40, 31 +(2003).
+
+
+
[53]
G. +Baskaran, S. Mandal, and R. Shankar, Exact Results for +Spin Dynamics and Fractionalization in the Kitaev Model, Phys. +Rev. Lett. 98, 247201 (2007).
+
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[54]
G. +Baskaran, D. Sen, and R. Shankar, Spin-S Kitaev Model: +Classical Ground States, Order from Disorder, and Exact Correlation +Functions, Phys. Rev. B 78, 115116 +(2008).
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[55]
Z. +Nussinov and G. Ortiz, Bond Algebras and +Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems, +Physical Review B 79, 214440 (2009).
+
+
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[56]
K. +O’Brien, M. Hermanns, and S. Trebst, Classification of +Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models, +Phys. Rev. B 93, 085101 (2016).
+
+
+
[57]
H. +Yao and S. A. Kivelson, An Exact Chiral +Spin Liquid with Non-Abelian Anyons, Phys. Rev. Lett. +99, 247203 (2007).
+
+
+
[58]
M. +Hermanns, K. O’Brien, and S. Trebst, Weyl Spin Liquids, +Physical Review Letters 114, 157202 (2015).
+
+
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[59]
T. +Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y. +Motome, and S. Trebst, Thermodynamics of a Gauge-Frustrated Kitaev +Spin Liquid, Physical Review Research 1, 032011(R) +(2019).
+
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V. +Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R. +Moessner, and R. Thomale, Non-Abelian Chiral +Spin Liquid on a Simple Non-Archimedean Lattice, Phys. Rev. B +101, 041114 (2020).
+
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Methods

The practical implementation of what is described in this section is available as a Python package called Koala (Kitaev On Amorphous