From a37e39ea5a5ea13ff3a61be9916e70b3ef2361d6 Mon Sep 17 00:00:00 2001 From: Tom Hodson Date: Fri, 7 Oct 2022 17:55:04 +0100 Subject: [PATCH] updates --- _thesis/0_Preface/0.1_Abstract.html | 60 +++++ _thesis/0_Preface/0.1_Declarations.html | 81 ++++++ _thesis/0_Preface/0.2_Aknowledgements.html | 72 ++++++ _thesis/0_Preface/0.2_Declarations.html | 80 ++++++ _thesis/0_Preface/0.3_Aknowledgements.html | 86 +++++++ _thesis/1_Introduction/1_Intro.html | 53 ++-- _thesis/2_Background/2.1_FK_Model.html | 50 ++-- _thesis/2_Background/2.2_HKM_Model.html | 50 ++-- _thesis/2_Background/2.4_Disorder.html | 36 ++- .../3.1_LRFK_Model.html | 44 ++-- .../3.2_LRFK_Methods.html | 16 +- .../3.3_LRFK_Results.html | 75 ++++-- .../4.1_AMK_Model.html | 57 +++-- .../4.2_AMK_Methods.html | 17 +- .../4.3_AMK_Results.html | 96 +++---- _thesis/5_Conclusion/5_Conclusion.html | 119 +++++---- .../6_Appendices/A.3_Lattice_Generation.html | 2 +- _thesis/6_Appendices/A.5_The_Projector.html | 21 +- _thesis/toc.html | 12 +- assets/thesis/amk_chapter/visual_kitaev_1.svg | 56 ++-- .../localisation_radius_vs_length.svg | 242 +++++++++--------- assets/thesis/intro_chapter/venn_diagram.svg | 233 +++++++++-------- assets/thesis/sandpile.png | Bin 0 -> 308672 bytes 23 files changed, 1029 insertions(+), 529 deletions(-) create mode 100644 _thesis/0_Preface/0.1_Abstract.html create mode 100644 _thesis/0_Preface/0.1_Declarations.html create mode 100644 _thesis/0_Preface/0.2_Aknowledgements.html create mode 100644 _thesis/0_Preface/0.2_Declarations.html create mode 100644 _thesis/0_Preface/0.3_Aknowledgements.html create mode 100644 assets/thesis/sandpile.png diff --git a/_thesis/0_Preface/0.1_Abstract.html b/_thesis/0_Preface/0.1_Abstract.html new file mode 100644 index 0000000..57db93e --- /dev/null +++ b/_thesis/0_Preface/0.1_Abstract.html @@ -0,0 +1,60 @@ +--- +title: Abstract +excerpt: +layout: none +image: + +--- + + + + + + + Abstract + + + + + + + + + + + + + + +{% capture tableOfContents %} +
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Large systems of interacting objects can give rise to a rich array of emergent behaviours. Make those objects quantum and the possibilities only expand. Interacting quantum many-body systems, as such systems are called, include essentially all physical systems. Luckily, we don’t usually need to consider this full quantum many-body description. The world at the human scale is essentially classical (not quantum), while at the microscopic scale of condensed matter physics we can often get by without interactions. Some systems, however, do require the full description. These are known as strongly correlated materials. Some of the most exciting topics in modern condensed matter fall under this umbrella: the spin liquids, the fractional quantum Hall effect, high temperature superconductivity and much more. Unfortunately, strongly correlated materials are notoriously difficult to study, defying many of the established theoretical techniques within the field. Enter exactly solvable models, these are interacting quantum many-body systems with extensively many local symmetries. The symmetries give rise to conserved charges which essentially break the model up into many non-interacting quantum systems which are more amenable to standard theoretical techniques. This thesis will focus on two such exactly solvable models.

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The first, the Falicov-Kimball (FK) model is an exactly solvable limit of the famous Hubbard model which describes itinerant fermions interacting with a classical Ising background field. Originally introduced to explain metal-insulator transitions, it has a rich set of ground state and thermodynamic phases. Disorder or interactions can turn metals into insulators and the FK model features both transitions. We will define a generalised FK model in 1D with long-range interactions. This model which shows a similarly rich phase diagram to its higher dimensional cousins. We use an exact Markov Chain Monte Carlo method to map the phase diagram and compute the energy resolved localisation properties of the fermions. This allows us to look at how the move to 1D affects the physics of the model. We show that the model can be understood by comparison to a simpler model of fermions coupled to binary disorder.

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The second, the Kitaev Honeycomb (KH) model, was the first solvable 2D model with a Quantum Spin Liquid (QSL) ground state. QSLs are generally expected to arise from Mott insulators when frustration prevents magnetic ordering all the way to zero temperature. The QSL state defies the traditional Landau-Ginzburg-Wilson paradigm of phases being defined by local order parameters. It is instead a topologically ordered phase. Recent work generalising non-interacting topological insulator phases to amorphous lattices raises the question of whether interacting phases like the QSLs can be similarly generalised. We extend the KH model to random lattices with fixed coordination number three generated by Voronoi partitions of the plane. We show that this model remains solvable and hosts a chiral amorphous QSL ground state. The presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian QSL phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a phase transition to a conducting thermal metal state and discuss possible experimental realisations.

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Versions of this document

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A PDF version of document is available online with either normal or double line spacing. An HTML version is also available with some figures animated.

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https://doi.org/10.5281/zenodo.7143205

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Statement of Originality

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I declare that the following work is entirely my own except where stated otherwise. Contributions from collaborators and others have been acknowledged by standard referencing practices. I have permissions to reproduce any third party copyrighted material which are included at the end of this thesis.

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I would like to thank my supervisor, Professor Johannes Knolle and co-supervisor Professor Derek Lee for guidance and support during this long process.

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Dan Hdidouan for being an example of how to weather the stress of a PhD with grace and kindness.

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Arnaud for help and guidance…

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Carolyn, Juraci, Ievgeniia and Loli for their patience and support.

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Nina del Ser

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Brian Tam for his endless energy on our many many calls while we served as joint Postgraduate reps for the department.

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All the students in CMTH, Halvard, Tom, Chris, Krishnan, David, Tonny, Emanuele … and particularly to Thank you to the CMT group at TUM in Munich, Alex and Rohit.

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Gino, Peru and Willian for their collaboration on the Amorphous Kitaev Model.

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Mr Jeffries who encouraged me to pursue physics

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All the gang from Munich, Toni, Mine, Mike, Claudi.

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Dan Simpson, the poet in residence at Imperial and one of my favourite collaborators during my time at Imperial.

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Lou Khalfaoui for keeping me sane during the lockdown of March 2022. Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design expertise and patience.

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All the I-Stemm team, Katerina, Jeremey, John, ….

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And finally, I’d like the thank the staff of the Camberwell Public Library where the majority of this thesis was written.

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We thank Angus MacKinnon for helpful discussions, Sophie Nadel for input when preparing the figures.

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Versions of this document

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A PDF version of document is available online with either normal or double line spacing. An HTML version is also available with some figures animated.

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+
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Statement of Originality

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I declare that the following work is entirely my own except where stated otherwise. Contributions from collaborators and others have been acknowledged by standard referencing practices. I have permissions to reproduce any third party copyrighted material which are included at the end of this thesis.

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Acknowledgements here.

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Interacting Quantum Many Body Systems

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When you take many objects and let them interact together, it is often easier to describe the behaviour of the group as a whole rather than the behaviour of the individual objects. Consider a flock of starlings like that of fig. 1. Watching the flock you’ll see that it has a distinct outline, that waves of density will sometimes propagate through the closely packed birds and that the flock seems to respond to predators as a distinct object. The natural description of this phenomenon is in terms of the flock, not the individual birds.

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The behaviours of the flock are an emergent phenomenon. The starlings are only interacting with their immediate six or seven neighbours  [1,2], what a physicist would call a local interaction. There is much philosophical debate about how exactly to define emergence  [3,4]. For our purposes, it is enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a radically different description from that of the individual objects.

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When you take many objects and let them interact together, when we describe the behaviours that arise it is often easier to talks in terms of the group rather than the behaviour of the individual objects. A flock of starlings like that of fig. 1 is a good example. If you were to sit and watch a flock like this, you’d see that it has a distinct outline, that waves of density will sometimes propagate through the closely packed birds and that the flock seems to respond to predators as a distinct object. The natural description of this phenomenon is in terms of the flock, not the individual birds.

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A flock is an emergent phenomenon. The starlings are only interacting with their immediate six or seven neighbours  [1,2], what a physicist would call a local interaction. There is much philosophical debate about how exactly to define emergence  [3,4]. For our purposes, it is enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a radically different description from that of the individual objects.

Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0 @@ -84,12 +90,12 @@ image:

To give an example closer to the topic at hand, our understanding of thermodynamics began with bulk properties like heat, energy, pressure and temperature  [5]. It was only later that we gained an understanding of how these properties emerge from microscopic interactions between very large numbers of particles  [6].

At its heart, condensed matter is the study of the behaviours that can emerge from large numbers of interacting quantum objects at low energy. When these three properties are present together (a large number of objects, those objects being quantum and the presence interactions between the objects), we call it an interacting quantum many body system. From these three ingredients, nature builds all manner of weird and wonderful materials.

Historically, we first made headway by ignoring interactions and quantum properties and looking at purely many-body systems. The ideal gas law and the Drude classical electron gas  [7] are good examples. Including interactions leads to the Ising model  [8], the Landau theory  [9] and the classical theory of phase transitions  [10]. In contrast, condensed matter theory got its start in quantum many-body theory where the only electron-electron interaction considered is the Pauli exclusion principle. Bloch’s theorem  [11], the core result of band theory, predicted the properties of non-interacting electrons in crystal lattices. It predicted, in particular, that band insulators arise when the electrons bands are filled, leaving the fermi level in a bandgap  [7]. In the same vein, advances were made in understanding the quantum origins of magnetism, including ferromagnetism and antiferromagnetism  [12].

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The development of Landau-Fermi Liquid theory explained why band theory works so well even where an analysis of the relevant energies suggests that it should not  [13]. Landau Fermi Liquid theory demonstrates that, in many cases where electron-electron interactions are significant, the system can still be described in terms of generalised non-interacting quasiparticles. This happens when the properties of the quasiparticles in the interacting system can be smoothly connected to the free fermions of the non-interacting system.

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However, there are systems where even Landau Fermi Liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations. They are thus called strongly correlated materials  [14]. The canonical examples are superconductivity  [15], the fractional quantum hall effect  [16] and the Mott insulators  [17,18]. We’ll start by looking at the latter but shall see that there are many links between the three topics.

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The development of Landau-Fermi liquid theory explained why band theory works so well even where an analysis of the relevant energies suggests that it should not  [13]. Landau Fermi Liquid theory demonstrates that, in many cases where electron-electron interactions are significant, the system can still be described in terms of generalised non-interacting quasiparticles. This happens when the properties of the quasiparticles in the interacting system can be smoothly connected to the free fermions of the non-interacting system.

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However, there are systems where even Landau-Fermi liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations. They are thus called strongly correlated materials  [14]. The canonical examples are superconductivity  [15], the fractional quantum hall effect  [16] and the Mott insulators  [17,18]. We’ll start by looking at the latter but shall see that there are many links between the three topics.

Mott Insulators

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Mott insulators (MIs) are remarkable because their electrical insulator properties come not from having filled bands but from electron-electron interactions other than Pauli exclusion. Electrical conductivity, the bulk movement of electrons, requires both that there are electronic states very close in energy to the ground state and that those states are delocalised so that they can contribute to macroscopic transport. Band insulators are systems whose Fermi level falls within a gap in the density of states and thus fail the first criteria. Band insulators derive their character from the characteristics of the underlying lattice. A third kind of insulator, the Anderson insulators, have only localised electronic states near the fermi level and therefore fail the second criteria. In a later section, I will discuss Anderson insulators and the disorder that drives them.

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Mott Insulators (MIs) are remarkable because their electrical insulator properties come not from having filled bands but from electron-electron interactions other than Pauli exclusion. Electrical conductivity, the bulk movement of electrons, requires both that there are electronic states very close in energy to the ground state and that those states are delocalised so that they can contribute to macroscopic transport. Band insulators are systems whose Fermi level falls within a gap in the density of states and thus fail the first criteria. Band insulators derive their character from the characteristics of the underlying lattice. A third kind of insulator, the Anderson insulators, have only localised electronic states near the fermi level and therefore fail the second criteria. In a later section, I will discuss Anderson insulators and the disorder that drives them.

Both band and Anderson insulators occur without electron-electron interactions. MIs, by contrast, require a many body picture to understand and thus elude band theory and single-particle methods.

Figure 2: Three key adjectives. Many Body: systems considered in the limit of large numbers of particles. Quantum: objects whose behaviour requires quantum mechanics to describe accurately. Interacting: the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to strongly correlated materials. @@ -100,11 +106,13 @@ image:

\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow} n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha},\]

where \(c^\dagger_{i\alpha}\) creates a spin \(\alpha\) electron at site \(i\) and the number operator \(n_{i\alpha}\) measures the number of electrons with spin \(\alpha\) at site \(i\). The sum runs over lattice neighbours \(\langle i,j \rangle\) including both \(\langle i,j \rangle\) and \(\langle j,i \rangle\) so that the model is Hermitian.

In the non-interacting limit \(U << t\), the model reduces to free fermions and the many-body ground state is a separable product of Bloch waves filled up to the Fermi level. On the other hand, the ground state in the interacting limit \(U >> t\) is a direct product of the local Hilbert spaces \(|0\rangle, |\uparrow\rangle, |\downarrow\rangle, |\uparrow\downarrow\rangle\). At half filling, one electron per site, each site becomes a local moment in the reduced Hilbert space \(|\uparrow\rangle, |\downarrow\rangle\) and thus acts like a spin-\(1/2\)  [26].

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The Mott insulating phase occurs at half filling \(\mu = \tfrac{U}{2}\). Here the model can be rewritten in a symmetric form \[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2}).\]

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The basic reason that the half filled state is insulating seems trivial. Any excitation must include states of double occupancy that cost energy \(U\). Hence, the system has a finite bandgap and is an interaction-driven MI. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this antiferromagnetic (AFM) order was actually the reason for the insulating behaviour. This would make sense since AFM order doubles the unit cell and can turn a system into a band insulator with an even number of electrons per unit cell  [27]. However, MIs have been found without magnetic order  [28,29]. Instead, the local moments may form a highly entangled state known as a quantum spin liquid, which will be discussed shortly.

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Various theoretical treatments of the Hubbard model have been made, including those based on Fermi liquid theory, mean field treatments, the local density approximation (LDA)  [30], dynamical mean-field theory  [31], density matrix renormalisation group methods  [32–34] and Markov chain Monte Carlo  [35–37]. None of these approaches are perfect. Strong correlations are poorly described by Fermi liquid theory and LDA approaches while mean field approximations do poorly in low dimensional systems. This theoretical difficulty has made the Hubbard model a target for cold atom simulations  [38].

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From here, the discussion will branch in two directions. First, I will discuss a limit of the Hubbard model called the Falicov-Kimball model. Second, I will look at quantum spin liquids and the Kitaev honeycomb model.

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The Falicov-Kimball Model

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The Mott insulating phase occurs at half filling \(\mu = \tfrac{U}{2}\). Here the model can be rewritten in a symmetric form

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\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2}).\]

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The basic reason that the half filled state is insulating seems trivial. Any excitation must include states of double occupancy that cost energy \(U\). Hence, the system has a finite bandgap and is an interaction-driven MI. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this Antiferromagnetic (AFM) order was actually the reason for the insulating behaviour. This would make sense since AFM order doubles the unit cell and can turn a system into a band insulator with an even number of electrons per unit cell  [27]. However, MIs have been found without magnetic order  [28,29]. Instead, the local moments may form a highly entangled state known as a Quantum Spin Liquid (QSL), which will be discussed shortly.

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Various theoretical treatments of the Hubbard model have been made, including those based on Fermi liquid theory, mean field treatments, the local density approximation  [30], dynamical mean-field theory  [31], density matrix renormalisation group methods  [32–34] and Markov chain Monte Carlo  [35–37]. None of these approaches are perfect. Strong correlations are poorly described by Fermi liquid theory and LDA approaches while mean field approximations do poorly in low dimensional systems. This theoretical difficulty has made the Hubbard model a target for cold atom simulations  [38].

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From here, the discussion will branch in two directions. First, I will discuss a limit of the Hubbard model called the Falicov-Kimball model. Second, I will look at QSLs and the Kitaev honeycomb model.

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The Falicov-Kimball Model

Figure 3: The Falicov-Kimball model can be viewed as a model of classical spins S_i coupled to spinless fermions \hat{c}_i where the fermions are mobile with hopping t and the fermions are coupled to the spins by an Ising type interaction with strength U. @@ -113,12 +121,13 @@ image:

\[\begin{aligned} H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}). \\ \end{aligned}\]

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The physics of states near the metal-insulator transition is still poorly understood  [40,41]. As a result, the FK model provides a rich test bed to explore interaction-driven metal-insulator transition physics. Despite its simplicity, the model has a rich phase diagram in \(D \geq 2\) dimensions. It shows a Mott insulator transition even at high temperature, similar to the corresponding Hubbard Model  [42]. In one dimension, the ground state phenomenology as a function of filling can be rich  [43], but the system is disordered for all \(T > 0\)  [44]. The model has also been a test-bed for many-body methods. Interest took off when an exact dynamical mean-field theory solution in the infinite dimensional case was found  [45–48].

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In chapter 3, I will introduce a generalized Falicov-Kimball model in one dimension I call the Long-Range Falicov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram like its higher dimensional cousins. Our goal is to understand the Mott transition in more detail, the phase transition into a charge density wave state and how the localisation properties of the fermionic sector behave in one dimension. We were particularly interested to see if correlations in the disorder potential are enough to bring about localisation effects, such as mobility edges, that are normally only seen in higher dimensions. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localisation properties of the fermions. We observe what appears to be a hint of coexisting localised and delocalised states. However, after careful comparison to an Anderson model of uncorrelated binary disorder about a background charge density wave field, we confirm that the fermionic sector does fully localise at larger system sizes as expected for one dimensional systems.

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The physics of states near the metal-insulator transition is still poorly understood  [40,41]. As a result, the FK model provides a rich test bed to explore interaction-driven metal-insulator transition physics. Despite its simplicity, the model has a rich phase diagram in \(D \geq 2\) dimensions. It shows a Mott insulator transition even at high temperature, similar to the corresponding Hubbard model  [42]. In 1D, the ground state phenomenology as a function of filling can be rich  [43], but the system is disordered for all \(T > 0\)  [44]. The model has also been a test-bed for many-body methods. Interest took off when an exact dynamical mean-field theory solution in the infinite dimensional case was found  [45–48].

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In chapter 3, I will introduce a generalised Falicov-Kimball model in 1D I call the Long-Range Falicov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram like its higher dimensional cousins. Our goal is to understand the Mott transition in more detail, the phase transition into a charge density wave state and how the localisation properties of the fermionic sector behave in 1D. We were particularly interested to see if correlations in the disorder potential are enough to bring about localisation effects, such as mobility edges, that are normally only seen in higher dimensions. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localisation properties of the fermions. We observe what appears to be a hint of coexisting localised and delocalised states. However, after careful comparison to an Anderson model of uncorrelated binary disorder about a background charge density wave field, we confirm that the fermionic sector does fully localise at larger system sizes as expected for 1D systems.

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Quantum Spin Liquids

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Turning to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long range order via a spontaneously broken symmetry. Within this paradigm, we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973  [49] that, if long range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature: a Quantum Spin Liquid (QSL).

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Turning to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long-range order via a spontaneously broken symmetry. Within this paradigm, we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973  [49] that, if long-range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature: a QSL.

This QSL state would exist at zero or very low temperatures. Therefore, we would expect quantum effects to be very strong, which will have far reaching consequences. It was the discovery of a different phase, however, that really kickstarted interest in the topic. The fractional quantum Hall state, discovered in the 1980s  [50] is an explicit example of an interacting electron system that falls outside of the Landau-Ginzburg-Wilson paradigm1. It shares many phenomenological properties with the QSL state. They both exhibit fractionalised excitations, braiding statistics and non-trivial topological properties  [55]. The many-body ground state of such systems acts as a complex and highly entangled vacuum. This vacuum can support quasiparticle excitations with properties unbound from that of the Dirac fermions of the standard model.

How do we actually make a QSL? Frustration is one mechanism that we can use to suppress magnetic order in spin models  [56]. Frustration can be geometric. Triangular lattices, for instance, cannot support AFM order. It can also come about as a result of spin-orbit coupling or other physics. There are also other routes to QSLs besides frustrated spin systems that we will not discuss here  [57–59].

@@ -128,15 +137,15 @@ H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U

Spin-orbit coupling is a relativistic effect that, very roughly, corresponds to the fact that in the frame of reference of a moving electron the electric field of nearby nuclei looks like a magnetic field to which the electron spin couples. This couples the spatial and spin parts of the electron wavefunction. The lattice structure can therefore influence the form of the spin-spin interactions, leading to spatial anisotropy in the effective interactions. This spatial anisotropy can frustrate an MI state  [60,61] leading to more exotic ground states than the AFM order we have seen so far. As with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral \(t\). Hence, we will see strong frustration in materials with strong spin-orbit coupling \(\lambda\) relative to their bandwidth \(t\).

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In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known as Kitaev materials, draw their name from the celebrated Kitaev Honeycomb model which is expected to model their low temperature behaviour  [56,62–65].

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In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known as Kitaev materials, draw their name from the celebrated Kitaev Honeycomb (KH) model which is expected to model their low temperature behaviour  [56,62–65].

At this point, we can sketch out a phase diagram like that of fig. 4. When both electron-electron interactions \(U\) and spin-orbit couplings \(\lambda\) are small relative to the bandwidth \(t\), we recover standard band theory of band insulators and metals. In the upper left, we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to topological insulators characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See  [66] for a more expansive version of this diagram.

-

The Kitaev Honeycomb model  [67] was one of the first exactly solvable spin models with a QSL ground state. It is defined on the two dimensional honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This 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, but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases  [68] and a finite temperature phase transition to a thermal metal state  [69]. It has been proposed that its non-Abelian excitations could be used to support robust topological quantum computing  [70–72].

-

The Kitaev model and FK model have quite a bit of conceptual overlap. They are both effectively models of spinless fermions coupled to a classical Ising background field. This is what makes them exactly solvable. At finite temperatures, fluctuations in their background fields provide an effective disorder potential for the fermionic sector, so both models can be studied at finite temperature with Markov chain Monte Carlo methods  [69,73].

-

As Kitaev points out in his original paper, the model remains solvable on any tri-coordinated \(z=3\) graph which can be 3-edge-coloured. Indeed many generalisations of the model exist  [74–78]. Notably, the Yao-Kivelson model  [79] introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry. This is likely because edge-colouring, finding the ground state and understanding the QSL properties are much harder without it  [80,81]. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev model. This would be a model of a tri-coordinated, highly bond anisotropic but otherwise amorphous material.

-

Amorphous materials do not have long-range lattice regularities but may have short-range regularities in the lattice structure, such as fixed coordination number \(z\) as in some covalent compounds. The best examples are amorphous Silicon and Germanium with \(z=4\) which are used to make thin-film solar cells  [82,83]. Recently, it has been shown that topological insulating phases can exist in amorphous systems. Amorphous topological insulators are characterised by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants  [84–90]. However, research on amorphous electronic systems has mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity  [91–95] in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs  [96].

+

The KH model  [67] was one of the first exactly solvable spin models with a QSL ground state. It is defined on the 2D honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This 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, but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases  [68] and a finite temperature phase transition to a thermal metal state  [69]. It has been proposed that its non-Abelian excitations could be used to support robust topological quantum computing  [70–72].

+

The KH and FK models have quite a bit of conceptual overlap. They can both be seen as models of spinless fermions coupled to a classical Ising background field. This is what makes them exactly solvable. At finite temperatures, fluctuations in their background fields provide an effective disorder potential for the fermionic sector, so both models can be studied at finite temperature with Markov chain Monte Carlo methods  [69,73].

+

As Kitaev points out in his original paper, the KH model remains solvable on any trivalent \(z=3\) graph which can be three-edge-coloured. Indeed many generalisations of the model exist  [74–78]. Notably, the Yao-Kivelson model  [79] introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry. This is likely because edge-colouring, finding the ground state and understanding the QSL properties are much harder without it  [80,81]. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev model. This would be a model of a trivalent, highly bond anisotropic but otherwise amorphous material.

+

Amorphous materials do not have long-range lattice regularities but may have short-range regularities in the lattice structure, such as fixed coordination number \(z\) as in some covalent compounds. The best examples are amorphous Silicon and Germanium with \(z=4\) which are used to make thin-film solar cells  [82,83]. Recently, it has been shown that topological insulating (TI) phases can exist in amorphous systems. Amorphous TIs are characterised by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants  [84–90]. However, research on amorphous electronic systems has mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity  [91–95] in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs  [96].

Amorphous magnetic systems have been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems  [97–100] and with numerical methods  [101,102]. Research on classical Heisenberg and Ising models accounts for the observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour  [103]. However, the role of spin-anisotropic interactions and quantum effects in amorphous magnets has not been addressed.

-

In chapter 4, I will address the question of whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL)  [104–107]. We will find that the answer is yes. I will introduce the Amorphous Kitaev (AK) model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. I will show that this model is a solvable, amorphous, chiral spin liquid. As with the Yao-Kivelson model  [79], the AK 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 will confirm prior observations that the form of the ground state is relatively simple  [77,108] and unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I will show that the system undergoes a finite-temperature phase transition to a thermal metal state and discuss possible experimental realisations.

-

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

+

In chapter 4, I will address the question of whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled QSL  [104–107]. We will find that the answer is yes. I will introduce the amorphous Kitaev model, a generalisation of the KH model to random lattices with fixed coordination number three. I will show that this model is a solvable, amorphous, chiral spin liquid. As with the Yao-Kivelson model  [79], the amorphous Kitaev 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 will confirm prior observations that the form of the ground state is relatively simple  [77,108] and unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I will show that the system undergoes a finite-temperature phase transition to a thermal metal state and discuss possible experimental realisations.

+

The next chapter, Chapter 2, will introduce some necessary background to the FK model, the KH model, and disorder and localisation.

Next Chapter: 2 Background

diff --git a/_thesis/2_Background/2.1_FK_Model.html b/_thesis/2_Background/2.1_FK_Model.html index 659743c..ce3d3ce 100644 --- a/_thesis/2_Background/2.1_FK_Model.html +++ b/_thesis/2_Background/2.1_FK_Model.html @@ -43,7 +43,7 @@ image:
  • Bibliography
    • @@ -63,7 +63,7 @@ image:
  • Bibliography
    • @@ -82,19 +82,19 @@ image:

    The Falicov Kimball Model

    The Model

    -

    The Falicov-Kimball (FK) model is one of the simplest models of the correlated electron problem. It captures the essence of the interaction between itinerant and localized electrons. It was originally introduced to explain the metal-insulator transition in f-electron systems but in its long history it has been interpreted variously as a model of electrons and ions, binary alloys or of crystal formation  [1–4]. In terms of immobile fermions \(d_i\) and light fermions \(c_i\) and with chemical potential fixed at half-filling, the model reads

    +

    The Falicov-Kimball (FK) model is one of the simplest models of the correlated electron problem. It captures the essence of the interaction between itinerant and localised electrons. It was originally introduced to explain the metal-insulator transition in f-electron systems but in its long history it has been interpreted variously as a model of electrons and ions, binary alloys or of crystal formation  [1–4]. In terms of immobile fermions \(d_i\) and light fermions \(c_i\) and with chemical potential fixed at half-filling, the model reads

    \[\begin{aligned} H_{\mathrm{FK}} = & \;U \sum_{i} (d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\ \end{aligned}\]

    Here we will only discuss the hypercubic lattices, i.e the chain, the square lattice, the cubic lattice and so on. The connection to the Hubbard model is that we have relabelled the up and down spin electron states and removed the hopping term for one spin state, the equivalent of taking the limit of infinite mass ratio  [5].

    -

    Like other exactly solvable models  [6], the FK model possesses extensively many conserved degrees of freedom \([d^\dagger_{i}d_{i}, H] = 0\). Similarly, the Kitaev Model model contains an extensive number of conserved fluxes. So in both models, the Hilbert space breaks up into a set of sectors in which these operators take a definite value. Crucially, this reduces the interaction terms in the model from being quartic in fermion operators to quadratic. This is what makes the two models exactly solvable, in contrast to the Hubbard model. For the FK model the interaction term \((d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\) becomes quadratic when \(d^\dagger_{i}d_{i}\) is replaced with on of its eigenvalues \(\{0,1\}\). The same thing happens in the Kitaev model, though after first applying a clever transformation which we will discuss later.

    +

    Like other exactly solvable models  [6], the FK model possesses extensively many conserved degrees of freedom \([d^\dagger_{i}d_{i}, H] = 0\). Similarly, the Kitaev model contains an extensive number of conserved fluxes. So in both models, the Hilbert space breaks up into a set of sectors in which these operators take a definite value. Crucially, this reduces the interaction terms in the model from being quartic in fermion operators to quadratic. This is what makes the two models exactly solvable, in contrast to the Hubbard model. For the FK model the interaction term \((d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\) becomes quadratic when \(d^\dagger_{i}d_{i}\) is replaced with on of its eigenvalues \(\{0,1\}\). The same thing happens in the Kitaev model, though after first applying a clever transformation which we will discuss later.

    Due to Pauli exclusion, maximum filling occurs when each lattice site is fully occupied, \(\langle n_c + n_d \rangle = 2\). Here we will focus on the half filled case \(\langle n_c + n_d \rangle = 1\). The ground state phenomenology as the model is doped away from the half-filled state can be rich  [7,8] but the half-filled point has symmetries that make it particularly interesting. From this point on we will only consider the half-filled point.

    -

    At half-filling and on bipartite lattices, the FK the model is particle-hole (PH) symmetric. That is, the Hamiltonian anticommutes with the particle hole operator \(\mathcal{P}H\mathcal{P}^{-1} = -H\). As a consequence, the energy spectrum is symmetric about \(E = 0\), which is the Fermi energy. The particle hole operator corresponds to the substitution \(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\) where \(\epsilon_i = +1\) for the A sublattice and \(-1\) for the B sublattice  [9]. The absence of a hopping term for the heavy electrons means they do not need the factor of \(\epsilon_i\) but they would need it in the corresponding Hubbard model. See appendix A.1 for a full derivation of the PH symmetry.

    +

    At half-filling and on bipartite lattices, the FK the model is particle-hole symmetric. That is, the Hamiltonian anticommutes with the particle hole operator \(\mathcal{P}H\mathcal{P}^{-1} = -H\). As a consequence, the energy spectrum is symmetric about \(E = 0\), which is the Fermi energy. The particle hole operator corresponds to the substitution \(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\) where \(\epsilon_i = +1\) for the A sublattice and \(-1\) for the B sublattice  [9]. The absence of a hopping term for the heavy electrons means they do not need the factor of \(\epsilon_i\) but they would need it in the corresponding Hubbard model. See appendix A.1 for a full derivation of the particle-hole symmetry.

    -Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With no external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black. - +Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in 1D. (a) With no external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black. +
    -

    We will later add a long range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field \(S_i = 1 - 2d^\dagger_id_i = \pm1\) which I will refer to as the spins.

    +

    We will later add a long-range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field \(S_i = 1 - 2d^\dagger_id_i = \pm1\) which I will refer to as the spins.

    \[\begin{aligned} H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\ \end{aligned}\]

    @@ -106,27 +106,27 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\ Figure 2: Schematic Phase diagram of the Falicov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from  [10,14] -

    In dimensions greater than one, the FK model exhibits a phase transition at some \(U\) dependent critical temperature \(T_c(U)\) to a low temperature ordered phase  [15]. In terms of the heavy electrons this corresponds to them occupying only one of the two sublattices A and B, known as a charge density wave (CDW) phase. In terms of spins this is an AFM phase.

    +

    In dimensions greater than one, the FK model exhibits a phase transition at some \(U\) dependent critical temperature \(T_c(U)\) to a low temperature ordered phase  [15]. In terms of the heavy electrons this corresponds to them occupying only one of the two sublattices A and B, known as a Charge Density Wave (CDW) phase. In terms of spins this is an antiferromagnetic phase.

    In the disordered region above \(T_c(U)\) there are two insulating phases. For weak interactions \(U << t\), thermal fluctuations in the spins act as an effective disorder potential for the fermions, causing them to localise and giving rise to an Anderson insulating (AI) phase  [16] which we will discuss more in section 2.3. For strong interactions \(U >> t\), the spins are not ordered but nevertheless their interaction with the electrons opens a gap, leading to a Mott insulator analogous to that of the Hubbard model  [17]. The presence of an interaction driven phase like the Mott insulator in an exactly solvable model is part of what makes the FK model such an interesting system.

    -

    By contrast, in the one dimensional FK model there is no finite-temperature phase transition (FTPT) to an ordered CDW phase  [18]. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In one dimension, disorder generally dominates: even the weakest disorder exponentially localises all single particle eigenstates. In the one dimensional FK model this means the whole spectrum is localised at all finite temperatures  [19–21]. Though at low temperatures the localisation length may be so large that the states appear extended in finite sized systems  [14]. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in one dimension  [22–24]

    -

    The absence of finite temperature ordered phases in one dimensional systems is a general feature. It can be understood as a consequence of the fact that domain walls are energetically cheap in one dimension. Thermodynamically, short-range interactions just cannot overcome the entropy of thermal defects in one dimension. However, the addition of longer range interactions can overcome this  [25,26].

    -

    However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions  [27–30] decays as \(r^{-1}\) in one dimension  [31]. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures  [25,32]. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at \(T = 0\) for the one dimensional FK model  [26].

    -

    The one dimensional FK model has been studied numerically, perturbatively in interaction strength \(U\) and in the continuum limit  [33]. The main results are that for attractive \(U > U_c\) the system forms electron spin bound state ‘atoms’ which repel on another  [34] and that the ground state phase diagram has a has a fractal structure as a function of electron filling, a devil’s staircase  [35,36].

    -

    Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter 3 we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long Range Ising Model which is the subject of the next section.

    +

    By contrast, in the 1D FK model there is no Finite-Temperature Phase Transition (FTPT) to an ordered CDW phase  [18]. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In 1D, disorder generally dominates: even the weakest disorder exponentially localises all single particle eigenstates. In the 1D FK model this means the whole spectrum is localised at all finite temperatures  [19–21]. Though at low temperatures the localisation length may be so large that the states appear extended in finite sized systems  [14]. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in 1D  [22–24]

    +

    The absence of finite temperature ordered phases in 1D systems is a general feature. It can be understood as a consequence of the fact that domain walls are energetically cheap in 1D. Thermodynamically, short-range interactions just cannot overcome the entropy of thermal defects in 1D. However, the addition of longer range interactions can overcome this  [25,26].

    +

    However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions  [27–30] decays as \(r^{-1}\) in 1D  [31]. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures  [25,32]. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at \(T = 0\) for the 1D FK model  [26].

    +

    The 1D FK model has been studied numerically, perturbatively in interaction strength \(U\) and in the continuum limit  [33]. The main results are that for attractive \(U > U_c\) the system forms electron spin bound state ‘atoms’ which repel on another  [34] and that the ground state phase diagram has a has a fractal structure as a function of electron filling, a devil’s staircase  [35,36].

    +

    Based on this primacy of dimensionality, we will go digging into the 1D case. In chapter 3 we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long-Range Ising (LRI) model which is the subject of the next section.

    -

    Long Ranged Ising model

    -

    The suppression of phase transitions is a common phenomena in one dimensional systems and the Ising model serves as the canonical illustration of this. In terms of classical spins \(S_i = \pm 1\) the standard Ising model reads

    +

    Long-Ranged Ising model

    +

    The suppression of phase transitions is a common phenomena in 1D systems and the Ising model serves as the canonical illustration of this. In terms of classical spins \(S_i = \pm 1\) the standard Ising model reads

    \[H_{\mathrm{I}} = \sum_{\langle ij \rangle} S_i S_j\]

    -

    Like the FK model, the Ising model shows an FTPT to an ordered state only in two dimensions and above. This can be understood via Peierls’ argument  [25,26] to be a consequence of the low energy penalty for domain walls in one dimensional systems.

    +

    Like the FK model, the Ising model shows an FTPT to an ordered state only in 2D and above. This can be understood via Peierls’ argument  [25,26] to be a consequence of the low energy penalty for domain walls in 1D systems.

    -Figure 3: Domain walls in the one dimensional Ising model cost finite energy because they affect only one interaction while in the long range Ising model it depends on how the interactions decay with distance. - +Figure 3: Domain walls in the 1D Ising model cost finite energy because they affect only one interaction while in the Long-Range Ising (LRI) model it depends on how the interactions decay with distance. +
    -

    Following Peierls’ argument, consider the difference in free energy \(\Delta F = \Delta E - T\Delta S\) between an ordered state and a state with single domain wall as in fig. 3. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size \(L\) we see that short range interactions produce a constant energy penalty \(\Delta E\) for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is \(\propto \ln L\). Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In two dimensions and above, the energy penalty of a large domain wall scales like \(L^{d-1}\) which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the one dimensional FK model.

    -

    In contrast the long range Ising (LRI) model \(H_{\mathrm{LRI}}\) can have an FTPT in one dimension.

    +

    Following Peierls’ argument, consider the difference in free energy \(\Delta F = \Delta E - T\Delta S\) between an ordered state and a state with single domain wall as in fig. 3. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size \(L\) we see that short range interactions produce a constant energy penalty \(\Delta E\) for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is \(\propto \ln L\). Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In 2D and above, the energy penalty of a large domain wall scales like \(L^{d-1}\) which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the 1D FK model.

    +

    In contrast the LRI model \(H_{\mathrm{LRI}}\) can have an FTPT in 1D.

    \[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|) S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]

    -

    Renormalisation group analyses show that the LRI model has an ordered phase in 1D for \(1 < \alpha < 2\)   [37]. Peierls’ argument can be extended  [32] to long range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state \(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\) and a domain wall state \(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\). In the case of the LRI model, careful counting shows that this energy penalty is \[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\qquad{(1)}\]

    +

    Renormalisation group analyses show that the LRI model has an ordered phase in 1D for \(1 < \alpha < 2\)   [37]. Peierls’ argument can be extended  [32] to long-range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state \(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\) and a domain wall state \(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\). In the case of the LRI model, careful counting shows that this energy penalty is \[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\qquad{(1)}\]

    because each interaction between spins separated across the domain by a bond length \(n\) can be drawn between \(n\) equivalent pairs of sites. The behaviour then depends crucially on how eq. 1 scales with system size. Ruelle proved rigorously for a very general class of 1D systems that if \(\Delta E\) or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic  [38]. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on \(J(n)\)  [37].

    With a power law form for \(J(n)\), there are a few cases to consider:

    For \(\alpha = 0\) i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact  [39]. This limit is the same as the infinite dimensional limit.

    @@ -135,10 +135,10 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\

    For \(\alpha = 2\), the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition  [32].

    Finally, for \(2 < \alpha\) we have very quickly decaying interactions and domain walls again have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.

    One final complexity is that for \(\tfrac{3}{2} < \alpha < 2\) renormalisation group methods show that the critical point has non-universal critical exponents that depend on \(\alpha\)   [43]. To avoid this potential confounding factors we will park ourselves at \(\alpha = 1.25\) when we apply these ideas to the FK model.

    -

    Were we to extend this to arbitrary dimension \(d\) we would find that thermodynamics properties generally both \(d\) and \(\alpha\), long range interactions can modify the ‘effective dimension’ of thermodynamic systems  [44].

    +

    Were we to extend this to arbitrary dimension \(d\) we would find that thermodynamics properties generally both \(d\) and \(\alpha\), long-range interactions can modify the ‘effective dimension’ of thermodynamic systems  [44].

    -Figure 4: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied. In my simulations I stick to a value of \alpha = \tfrac{5}{4} the complexity of non-universal critical exponents. - +Figure 4: The thermodynamic behaviour of the long-range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied. In my simulations I stick to a value of \alpha = \tfrac{5}{4} the complexity of non-universal critical exponents. +

    Next Section: The Kitaev Honeycomb Model

    diff --git a/_thesis/2_Background/2.2_HKM_Model.html b/_thesis/2_Background/2.2_HKM_Model.html index 160bc61..d4fcb4c 100644 --- a/_thesis/2_Background/2.2_HKM_Model.html +++ b/_thesis/2_Background/2.2_HKM_Model.html @@ -89,31 +89,31 @@ image:

    The Spin Hamiltonian

    -

    This section introduces the Kitaev honeycomb (KH) model. The KH model is an exactly solvable model of interacting spin\(-1/2\) spins on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a label \(\alpha \in \{ x, y, z\}\) and that bond couple two spins along the \(\alpha\) axis. See fig. 1 for a diagram of the setup.

    +

    The Kitaev Honeycomb (KH) model is an exactly solvable model of interacting spin\(-1/2\) spins on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a label \(\alpha \in \{ x, y, z\}\) and couples two spins along the \(\alpha\) axis. See fig. 1 for a diagram of the setup.

    This gives us the Hamiltonian \[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha}, \qquad{(1)}\] where \(\sigma^\alpha_j\) is the \(\alpha\) component of a Pauli matrix acting on site \(j\) and \(\langle j,k\rangle_\alpha\) is a pair of nearest-neighbour indices connected by an \(\alpha\)-bond with exchange coupling \(J^\alpha\). Kitaev introduced this model in his seminal 2006 paper  [1].

    -

    The KH can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds  [2–6]. The model is highly frustrated: each spin would like to align along a different direction with each of its three neighbours but this cannot be achieved even classically  [7,8]. This frustration leads the model to have a quantum spin liquid (QSL) ground state, a complex many body state with a high degree of entanglement but no long range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier  [9], the KH model was the first exactly solvable models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations  [10]. At finite temperature the KH model undergoes a phase transition to a thermal metal state  [11]. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field with the remaining fermions are governed by a free fermion hamiltonian.

    +

    The KH can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds  [2–6]. The model is highly frustrated: each spin would like to align along a different direction with each of its three neighbours but this cannot be achieved even classically  [7,8]. This frustration leads the model to have a Quantum Sping Liquid (QSL) ground state, a complex many-body state with a high degree of entanglement but no long-range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier  [9], the KH model was the first exactly solvable models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations  [10]. At finite temperature the KH model undergoes a phase transition to a thermal metal state  [11]. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field with the remaining fermions are governed by a free fermion hamiltonian.

    This section will go over the standard model in detail, first discussing the spin model, then detailing the transformation to a Majorana hamiltonian that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the emergent gauge fields and the projector. The next section will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics. Finally will then discuss the ground state found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the phase diagram.

    The Spin Model

    -Figure 2: A visual introduction to the Kitaev Model. - +Figure 2: A visual introduction to the Kitaev honeycomb model. +
    -

    As discussed in the introduction, spin hamiltonians like that of the Kitaev model arise in electronic systems as the result the balance of multiple effects  [5]. For instance, in certain transition metal systems with \(d^5\) valence electrons, crystal field and spin-orbit couplings conspire to shift and split the \(d\) orbitals into moments with spin \(j = 1/2\) and \(j = 3/2\). Of these, the bandwidth \(t\) of the \(j= 1/2\) band is small, meaning that even relatively meagre electron correlations (such those induced by the \(U\) term in the Hubbard model) can lead to the opening of a Mott gap. From there we have a \(j = 1/2\) Mott insulator whose effective spin-spin interactions are again shaped by the lattice geometry and spin-orbit coupling leading some materials to have strong bond-directional Ising-type interactions  [12,13]. In the Kitaev Model the bond directionality refers to the fact that the coupling axis \(\alpha\) in terms like \(\sigma_j^{\alpha}\sigma_k^{\alpha}\) is strongly bond dependent.

    +

    As discussed in the introduction, spin hamiltonians like that of the KH model arise in electronic systems as the result the balance of multiple effects  [5]. For instance, in certain transition metal systems with \(d^5\) valence electrons, crystal field and spin-orbit couplings conspire to shift and split the \(d\) orbitals into moments with spin \(j = 1/2\) and \(j = 3/2\). Of these, the bandwidth \(t\) of the \(j= 1/2\) band is small, meaning that even relatively meagre electron correlations (such those induced by the \(U\) term in the Hubbard model) can lead to the opening of a Mott gap. From there we have a \(j = 1/2\) Mott insulator whose effective spin-spin interactions are again shaped by the lattice geometry and spin-orbit coupling leading some materials to have strong bond-directional Ising-type interactions  [12,13]. In the KH model the bond directionality refers to the fact that the coupling axis \(\alpha\) in terms like \(\sigma_j^{\alpha}\sigma_k^{\alpha}\) is strongly bond dependent.

    In the spin hamiltonian eq. 1 we can already tease out a set of conserved fluxes that will be key to the model’s solution. These fluxes are the expectations of Wilson loop operators

    \[\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha},\]

    the products of bonds winding around a closed path \(p\) on the lattice. These operators commute with the Hamiltonian and so have no time dynamics. The winding direction does not matter so long as it is fixed. By convention we will always use clockwise. Each closed path on the lattice is associated with a flux. The number of conserved quantities grows linearly with system size and is thus extensive, this is a common property for exactly solvable systems and can be compared to the heavy electrons present in the Falicov-Kimball model. The square of two loop operators is one so any contractible loop can be expressed as a product of loops around plaquettes of the lattice, as in fig. 3. For the honeycomb lattice the plaquettes are the hexagons. The expectations of \(\hat{W}_p\) through each plaquette, the fluxes, are therefore enough to describe the whole flux sector. We will focus on these fluxes, denoting them by \(\phi_i\). Once we have made the mapping to the Majorana Hamiltonian I will explain how these fluxes can be connected to an emergent \(B\) field which makes their interpretation as fluxes clear.

    -Figure 3: In the Kitaev Honeycomb model, Wilson loop operators \hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha} can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \phi_i. This relationship between the u_{ij} around a region and fluxes with one is evocative of Stokes’ theorem for classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later. - +Figure 3: In the Kitaev honeycomb model, Wilson loop operators \hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha} can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \phi_i. This relationship between the u_{ij} around a region and fluxes with one is evocative of Stokes’ theorem from classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later. +
    -

    It is worth noting in passing that the effective Hamiltonian for many Kitaev materials incorporates a contribution from an isotropic Heisenberg term \(\sum_{i,j} \vec{\sigma}_i\cdot\vec{\sigma}_j\), this is referred to as the Heisenberg-Kitaev Model  [14]. Materials for which the Kitaev term dominates are generally known as Kitaev Materials. See  [5] for a full discussion of Kitaev Materials.

    -

    As with the Falicov-Kimball model, the KH model has a extensive number of conserved quantities, the fluxes. As with the FK model it will make sense to work in the simultaneous eigenbasis of the fluxes and the Hamiltonian so that we can treat the fluxes like a classical degree of freedom. This is part of what makes the model tractable. We will find that the ground state of the model corresponds to some particular choice of fluxes. We will refer to local excitations away from the flux ground state as vortices. In order to fully solve the model however, we must first move to a Majorana picture.

    +

    It is worth noting in passing that the effective Hamiltonian for many Kitaev materials incorporates a contribution from an isotropic Heisenberg term \(\sum_{i,j} \vec{\sigma}_i\cdot\vec{\sigma}_j\), this is referred to as the Heisenberg-Kitaev model  [14]. Materials for which the Kitaev term dominates are generally known as Kitaev Materials. See  [5] for a full discussion of Kitaev Materials.

    +

    As with the Falicov-Kimball model, the KH model has a extensive number of conserved quantities, the fluxes. So again we will work in the simultaneous eigenbasis of the fluxes and the Hamiltonian so that we can treat the fluxes like a classical degree of freedom. This is part of what makes the model tractable. We will find that the ground state of the model corresponds to some particular choice of fluxes. We will refer to local excitations away from the flux ground state as vortices. In order to fully solve the model however, we must first move to a Majorana picture.

    The Majorana Model

    -

    Majorana fermions are something like ‘half of a complex fermion’ and are their own antiparticle. From a set of \(N\) fermionic creation \(f_i^\dagger\) and anhilation \(f_i\) operators we can construct \(2N\) Majorana operators \(c_m\). We can do this construction in multiple ways subject to only mild constraints required to keep the overall commutations relations correct  [1]. Majorana operators square to one but otherwise have standard fermionic commutation relations.

    +

    Majorana fermions are something like ‘half of a complex fermion’ and are their own antiparticle. From a set of \(N\) fermionic creation \(f_i^\dagger\) and anhilation \(f_i\) operators we can construct \(2N\) Majorana operators \(c_m\). We can do this construction in multiple ways subject to only mild constraints required to keep the overall commutations relations correct  [1]. Majorana operators square to one but otherwise have standard fermionic anti-commutation relations.

    \(N\) spins can be mapped to \(N\) fermions with the well known Jordan-Wigner transformation and indeed this approach can be used to solve the Kitaev model  [15]. Here I will introduce the method Kitaev used in the original paper as this forms the basis for the results that will be presented in this thesis. Rather than mapping to \(N\) fermions, Kitaev maps to \(4N\) Majoranas, effectively \(2N\) fermions. In contrast to the Jordan-Wigner approach which makes fermions out of strings of spin operators in order to correctly produce fermionic commutation relations, the Kitaev transformation maps each spin locally to four Majoranas. The downside is that this enlarges the Hilbert space from \(2^N\) to \(4^N\). We will have to employ a projector \(\hat{P}\) to come back down to the physical Hilbert space later. As everything is local, I will drop the site indices \(ijk\) in expressions that refer to only a single site.

    The mapping is defined in terms of four Majoranas per site \(b_i^x,\;b_i^y,\;b_i^z,\;c_i\) such that

    \[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^z = i b^z c\qquad{(2)}\]

    @@ -126,7 +126,7 @@ b^x = (f + f^\dagger),\;\;& b^y = -i(f - f^\dagger),\\ b^z = (g + g^\dagger),\;\;& c = -i(g - g^\dagger), \end{aligned}\]

    Working through the algebra we see that the operator \(D = b^x b^y b^z c\) is equal to the fermion parity \(D = -(2n_f - 1)(2n_g - 1) = \pm1\) where \(n_f,\; n_g\) are the number operators. So setting \(D = 1\) everywhere is equivalent to restricting to the \(\{|01\rangle,|10\rangle\}\) though we could equally well have used the other one.

    -

    Expanding the product \(\prod_i P_i\) out, we find that the projector corresponds to a symmetrisation over \(\{u_{ij}\}\) states within a flux sector and and overall fermion parity \(\prod_i D_i\). The significance of this is that an arbitrary many body state can be made to have non-zero overlap with the physical subspace via the addition or removal of just a single fermion. This implies that in the thermodynamic limit the projection step is not generally necessary to extract physical results, see  [17] or appendix A.5 for more details.

    +

    Expanding the product \(\prod_i P_i\) out, we find that the projector corresponds to a symmetrisation over \(\{u_{ij}\}\) states within a flux sector and and overall fermion parity \(\prod_i D_i\), see  [17] or appendix A.5 for the full derivation. The significance of this is that an arbitrary many-body state can be made to have non-zero overlap with the physical subspace via the addition or removal of just a single fermion. This implies that in the thermodynamic limit the projection step is not generally necessary to extract physical results

    We can now rewrite the spin hamiltonian in Majorana form with the caveat that they are only strictly equivalent after projection. The Ising interactions \(\sigma_j^{\alpha}\sigma_k^{\alpha}\) decouple into the form \(-i (i b^\alpha_i b^\alpha_j) c_i c_j\). We factor out the bond operators \(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\) which are Hermitian and, remarkably, commute with the Hamiltonian and each other.

    \[\begin{aligned} \tilde{H} &= - \sum_{\langle i,j\rangle_\alpha} J^{\alpha}\tilde{\sigma}_i^{\alpha}\tilde{\sigma}_j^{\alpha}\\ @@ -148,7 +148,7 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}

    \[ f_i = \tfrac{1}{2} (b_m + ib_m')\]

    with their associated number operators \(n_i = f^\dagger_i f_i\). These let us write the Hamiltonian neatly as

    \[ H = \sum_m \epsilon_m (n_m - \tfrac{1}{2}).\]

    -

    The energy of the ground state \(|n_m = 0\rangle\) of the many body system at fixed \(\{u_{ij}\}\) is

    +

    The energy of the ground state \(|n_m = 0\rangle\) of the many-body system at fixed \(\{u_{ij}\}\) is

    \[E_{0} = -\frac{1}{2}\sum_m \epsilon_m \]

    and we can construct any state from a particular choice of \(n_m = 0,1\). If we only care about the ground state energy \(E_{0}\), it is possible to skip forming the fermionic operators. The eigenvalues obtained directly from diagonalising \(J^{\alpha} u_{ij}\) come in \(\pm \epsilon_m\) pairs. We can take half the absolute value of the set to recover \(\sum_m \epsilon_m\) directly.

    @@ -157,11 +157,11 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}

    We have transformed the spin Hamiltonian into a Majorana hamiltonian \(H = i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\) describing the dynamics of a classical field \(u_{ij}\) and Majoranas \(c_i\). It is natural to ask how the classical field \(u_{ij}\) relates to the fluxes of the original spin model. We can evaluate the fluxes \(\phi_i\) in terms of the bond operators

    \[\phi_i = \prod_{\langle j,k\rangle \in \mathcal{P}_i} i u_{jk}.\qquad{(4)}\]

    -Figure 4: A honeycomb lattice with edges in light grey, along with its dual, the triangle lattice in light blue. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator D_j at a vertex is to flip the value of the three u_{jk} variables (black lines) surrounding site j. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent D_j operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing D_j operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of u_{ij}s flipped on its edge. Therefore, all retain the same flux \phi_i. - +Figure 4: A honeycomb lattice with edges in grey, along with its dual, the triangle lattice in red. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator D_j at a vertex is to flip the value of the three u_{jk} variables (black lines) surrounding site j. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent D_j operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing D_j operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of u_{ij}s flipped on its edge. Therefore, all retain the same flux \phi_i. +

    In addition, the bond operators form a highly degenerate description of the system. The operators \(D_i = b^x_i b^y_i b^z_i c_i\) commute with \(H\) forming a set of local symmetries. The action of \(D_i\) on a state is to flip the values of the three \(u_{ij}\) bonds that connect to site \(i\). This changes the bond configuration \(\{u_{ij}\}\) but leaves the flux configuration \(\{\phi_i\}\) unchanged. Physically, we interpret \(u_{ij}\) as a gauge field with a high degree of degeneracy and \(\{D_i\}\) as the set of gauge symmetries. The Majorana bond operators \(u_{ij}\) are an emergent, classical, \(\mathbb{Z}_2\) gauge field! The flux configuration \(\{\phi_i\}\) is what encodes physical information about the system without all the gauge degeneracy.

    -

    The ground state of the Kitaev Honeycomb model is the all one flux configuration \(\{\phi_i = +1\; \forall \; i\}\). This can be proven via Lieb’s theorem  [19] which gives the lowest energy magnetic flux configuration for a system of electrons hopping in a magnetic field. Kitaev remarks in his original paper that he was not initially aware of the relevance of Lieb’s 1994 result. This is not surprising because at first glance the two models seem quite different but the connection is quite instructive for understanding the Kitaev Model and its generalisations.

    +

    The ground state of the KH model is the flux configuration where all fluxes are one \(\{\phi_i = +1\; \forall \; i\}\). This can be proven via Lieb’s theorem  [19] which gives the lowest energy magnetic flux configuration for a system of electrons hopping in a magnetic field. Kitaev remarks in his original paper that he was not initially aware of the relevance of Lieb’s 1994 result. This is not surprising because at first glance the two models seem quite different but the connection is quite instructive for understanding the KH and its generalisations.

    Lieb discusses a model of mobile electrons

    \[H = \sum_{ij} t_{ij} c^\dagger_i c_j\]

    where the hopping terms \(t_{ij} = |t_{ij}|\exp(i\theta_{ij})\) incorporate Aharanhov-Bohm (AB) phases  [20] \(\theta_{ij}\). The AB phases model the effect of a slowly varying magnetic field on the electrons through the integral of the magnetic vector potential \(\theta_{ij} = \int_i^j \vec{A} \cdot d\vec{l}\), a Peierls substitution  [21]. If we map the Majorana form of the Kitaev model to Lieb’s model we see that our \(t_{ij} = i J^\alpha u_{ij}\). The \(i u_{ij} = \pm i\) correspond to AB phases \(\theta_{ij} = \pi/2\) or \(3\pi/2\) along each bond.

    @@ -179,7 +179,7 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c} \end{aligned}\qquad{(5)}\]

    Thus we can interpret the fluxes \(\phi_i\) as the exponential of magnetic fluxes \(Q_m\) of some fictitious gauge field \(\vec{A}\) and the bond operators as \(i u_{ij} = \exp i \int_i^j \vec{A} \cdot d\vec{l}\). In this analogy to classical electromagnetism, the sets \(\{u_{ij}\}\) that correspond to the same \(\{\phi_i\}\) are all gauge equivalent as we have already seen via other means. The fact that fluxes can be written as products of bond operators and composed is a consequence of eq. 5. If the lattice contains odd plaquettes, as in the Yao-Kivelson model  [26], the complex fluxes that appear are a sign that chiral symmetry has been broken.

    In full, Lieb’s theorem states that the ground state has magnetic flux \(Q_i = \sum_{\mathcal{P}_i}\theta_{ij} = \pi \; (\mathrm{mod} \;2\pi)\) for plaquettes with \(0 \; (\mathrm{mod}\;4)\) sides and \(0 \; (\mathrm{mod}\;2\pi)\) for plaquettes with \(2 \; (\mathrm{mod}\;4)\) sides. In terms of our fluxes, this means \(\phi = -1\) for squares, \(\phi = 1\) for hexagons and so on.

    -

    While Lieb’s theorem is restricted to bipartite lattices with translational symmetry, other works has shown numerically that it tends to hold for more general lattices too  [22–25]. From this we find that the generalisation to odd sided plaquettes is similar but with an additional chiral symmetry, so \(\phi = \pm i\) for plaquettes with \(1 \; (\mathrm{mod}\;4)\) sides and \(\mp i\) for those with \(3 \; (\mathrm{mod}\;4)\) sides. Overall we can write \(\phi = -(\pm i)^{n_{\mathrm{sides}}}\). Later I will present numerical evidence that this rule continues to hold for general amorphous lattices.

    +

    While Lieb’s theorem is restricted to bipartite lattices with translational symmetry, other works have shown numerically that it tends to hold for more general lattices too  [22–25]. From this we find that the generalisation to odd sided plaquettes is similar but with an additional chiral symmetry, so \(\phi = \pm i\) for plaquettes with \(1 \; (\mathrm{mod}\;4)\) sides and \(\mp i\) for those with \(3 \; (\mathrm{mod}\;4)\) sides. Overall we can write \(\phi = -(\pm i)^{n_{\mathrm{sides}}}\). Later I will present numerical evidence that this rule continues to hold for general amorphous lattices.

    Understanding \(u_{ij}\) as a gauge field provides another way to understand the action of the projector. The local projector \(P_i = \frac{1 + D_i}{2}\) applied to a state constructs a superposition of the original state and the gauge equivalent state linked to it by flipping the three \(u_{ij}\) around site \(i\). The overall projector \(P = \prod_i P_i\) can thus be understood as a symmetrisation over all gauge equivalent states, removing the gauge degeneracy introduced by the mapping from spins to Majoranas.

    The Euler Equation

    -

    Euler’s equation provides a convenient way to understand how the states of the AK model factorise into flux sectors, gauge sectors and topological sectors. The Euler equation states if we embed a lattice with \(B\) bonds, \(P\) plaquettes and \(V\) vertices onto a closed surface of genus \(g\), (\(0\) for the sphere, \(1\) for the torus) then

    +

    Euler’s equation provides a convenient way to understand how the states of the AK model factorise into flux sectors, gauge sectors and topological sectors as in fig. 2. The Euler equation states that if we embed a lattice with \(B\) bonds, \(P\) plaquettes and \(V\) vertices onto a closed surface of genus \(g\), (\(0\) for the sphere, \(1\) for the torus) then

    \[B = P + V + 2 - 2g\]

    For the case of the torus where \(g = 1\), we can rearrange this and exponentiate it to read:

    \[2^B = 2^{P-1}\cdot 2^{V-1} \cdot 2^2\]

    There are \(2^B\) configurations of the bond variables \(\{u_{ij}\}\). Each of these configurations can be uniquely decomposed into a flux sector, a gauge sector and a topological sector, see fig. 2. Each of the \(P\) plaquette operators \(\phi_i\) takes two values but vortices are created in pairs so there are \(2^{P-1}\) vortex sectors in total. There are \(2^{V-1}\) gauge symmetries formed from the \(V\) symmetry operators \(D_i\) because \(\prod_{j} D_j = \mathbb{I}\) is enforced by the projector. Finally, the two topological fluxes \(\Phi_x\) and \(\Phi_y\) account for the last factor of \(2^2\).

    -

    In addition, the fact that we only work with trivalent lattices implies that each vertex shares three bonds with other vertices so effectively comes with \(\tfrac{3}{2}\) bonds. This is consistent with the fact that, in the Majorana representation on the torus, each vertex brings three \(b^\alpha\) operators which then pair along bonds to give \(3/2\) bonds per vertex. Substituting \(3V = 2B\) into Euler’s equation tells us that any trivalent lattice on the torus with \(N\) plaquettes has \(2N\) vertices and \(3N\) bonds. Since each bond is part of two plaquettes this implies that the mean number of sides of a plaquette is exactly six and that odd sides plaquettes must come in pairs.

    +

    In addition, the fact that we only work with trivalent lattices implies that each vertex shares three bonds with other vertices so effectively comes with \(3/2\) bonds. This is consistent with the fact that, in the Majorana representation on the torus, each vertex brings three \(b^\alpha\) operators which then pair along bonds to give \(3/2\) bonds per vertex. Substituting \(3V = 2B\) into Euler’s equation tells us that any trivalent lattice on the torus with \(N\) plaquettes has \(2N\) vertices and \(3N\) bonds. Since each bond is part of two plaquettes this implies that the mean number of sides of a plaquette is exactly six and that odd sides plaquettes must come in pairs.

    Next Section: Methods

    diff --git a/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html b/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html index e744130..e8537d9 100644 --- a/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html +++ b/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html @@ -76,15 +76,14 @@ image:

    Methods

    -

    This section describes the novel methods we developed to simulate the AK model including lattice generation, bond colouring and the inverse mapping between flux sector and gauge sector. Implementations are available online as a Python package called Koala (Kitaev On Amorphous LAttices)  [1]. All results and figures herein were generated with Koala.

    +

    This section describes the novel methods we developed to simulate the AK model including lattice generation, bond colouring and the inverse mapping between flux sector and gauge sector. All results and figures herein were generated with Koala  [1].

    Voronisation

    Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from \mathbb{R}^2. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus.
    -

    The lattices we use are Voronoi partitions of the torus  [2–4]. We start by sampling seed points uniformly (or otherwise) on the torus. As most off the shelf routines for computing Voronoi partitions are defined on the plane rather than the torus, we tile our seed points into a \(3\times3\) pr \(5\times5\) grid before calling a standard Voronoi routine  [5] from the python package Scipy  [6]. Finally, we undo the tiling to the grid by identifying edges in the tiled lattice which are identical, yielding a trivalent lattice on the torus. We encode our lattices with edge lists \([(i,j), (j,k)\ldots]\) and an additional vector \((\{-1,0,+1\}, \{-1,0,+1\})\) for each edge that encodes the sense in which it crosses the periodic boundary conditions, equivalent to how the edge leaves the unit cell were the system to tile the plane, see appendix A.3 for more detail.

    -

    The graph generated by a Voronoi partition of a two dimensional surface is always planar. This means that no edges cross each other when the graph is embedded into the plane. It is also trivalent in that every vertex is connected to exactly three edges  [7,8].

    +

    The lattices we use are Voronoi partitions of the torus  [2–4]. We start by sampling seed points uniformly on the torus. As most off the shelf routines for computing Voronoi partitions are defined on the plane rather than the torus, we tile our seed points into a \(3\times3\) pr \(5\times5\) grid before calling a standard Voronoi routine  [5] from the python package Scipy  [6]. Finally, we undo the tiling to the grid by identifying edges in the tiled lattice which are identical, yielding a trivalent lattice on the torus. We encode our lattices with edge lists \([(i,j), (k,l)\ldots]\) and an additional 2D vector \(\vec{v} \in \{-1,0,+1\}^2\) for each edge that encodes the sense in which it crosses the periodic boundary conditions. This is equivalent to how the edge would leave the unit cell were the system to tile the plane, see appendix A.3 for more detail. The graph generated by a Voronoi partition of a 2D surface is always planar. This means that no edges cross each other when the graph is embedded into the plane. It is also trivalent in that every vertex is connected to exactly three edges  [7,8].

    Colouring the Bonds

    @@ -92,11 +91,11 @@ image: Figure 2: Different valid three-edge-colourings of an amorphous lattice. Colors that differ from the leftmost panel are highlighted in the other panels. -

    To be solvable the AK model requires that each edge in the lattice be assigned a label \(x\), \(y\) or \(z\), such that each vertex has exactly one edge of each type connected to it. This problem must be distinguished from that considered by the famous four-colour theorem  [9]. The four-colour theorem is concerned with assigning colours to the vertices of planar graphs, such that no vertices that share an edge have the same colour. Here we are instead concerned with finding an edge colouring.

    -

    For a graph of maximum degree \(\Delta\), \(\Delta + 1\) colours are always enough to edge-colour it. An \(\mathcal{O}(mn)\) algorithm exists to do this for a graph with \(m\) edges and \(n\) vertices  [10]. Restricting ourselves to graphs with \(\Delta = 3\), these graphs are known as cubic graphs. Cubic graphs can be four-edge-coloured in linear time  [11]. However we need a three-edge-colouring of our cubic graphs, which turns out to be more difficult. Cubic, planar, bridgeless graphs can be three-edge-coloured if and only if they can be four-face-coloured  [12]. Bridges are edges that connect otherwise disconnected components. An \(\mathcal{O}(n^2)\) algorithm exists for these  [13]. However, it is not clear whether this extends to cubic, toroidal bridgeless graphs.

    +

    To be solvable, the AK model requires that each edge in the lattice be assigned a label \(x\), \(y\) or \(z\), such that each vertex has exactly one edge of each type connected to it, a three-edge-colouring. This problem must be distinguished from that considered by the famous four-colour theorem  [9]. The four-colour theorem is concerned with assigning colours to the vertices of planar graphs, such that no vertices that share an edge have the same colour.

    +

    For a graph of maximum degree \(\Delta\), \(\Delta + 1\) colours are always enough to edge-colour it. An \(\mathcal{O}(mn)\) algorithm exists to do this for a graph with \(m\) edges and \(n\) vertices  [10]. Graphs with \(\Delta = 3\) are known as cubic graphs. Cubic graphs can be four-edge-coloured in linear time  [11]. However we need a three-edge-colouring of our cubic graphs, which turns out to be more difficult. Cubic, planar, bridgeless graphs can be three-edge-coloured if and only if they can be four-face-coloured  [12]. Bridges are edges that connect otherwise disconnected components. An \(\mathcal{O}(n^2)\) algorithm exists for these  [13]. However, it is not clear whether this extends to cubic, toroidal bridgeless graphs.

    A four-face-colouring is equivalent to a four-vertex-colouring of the dual graph, see appendix A.3. So if we could find a four-vertex-colouring of the dual graph we would be done. However vertex-colouring a toroidal graph may require up to seven colours  [14]! The complete graph of seven vertices \(K_7\) is a good example of a toroidal graph that requires seven colours.

    -

    Luckily, some problems are harder in theory than in practice. Three-edge-colouring cubic toroidal graphs appears to be one of those things. To find colourings, we use a Boolean Satisfiability Solver or SAT solver. A SAT problem is a set of statements about a set of boolean variables, such as “\(x_1\) or not \(x_3\) is true”. A solution to a SAT problem is a assignment \(x_i \in {0,1}\) that satisfies all the statements  [15]. General purpose, high performance programs for solving SAT problems have been an area of active research for decades  [16]. Such programs are useful because, by the Cook-Levin theorem, any NP problem can be encoded (in polynomial time) as an instance of a SAT problem . This property is what makes SAT one of the subset of NP problems called NP-Complete  [17,18]. Thus, it is a relatively standard technique in the computer science community to solve NP problems by first transforming them to SAT instances and then using an off the shelf SAT solver. The output of this can then be mapped back to the original problem domain.

    -

    Whether graph colouring problems are in NP or P seems to depend delicately on the class of graphs considered, the maximum degree and the number of colours used. Since we I didn’t know of any better algorithm for the problem at hand using a SAT solver appeared to be a reasonable first method to try and it turns out to be fast enough in practice that it is by no means to rate limiting step for solving instances of our model. In appendix A.3 I detail the specifics of how I mapped edge-colouring problems to SAT instances and show a breakdown of where the computational effort is spent, the majority being on diagonalisation.

    +

    Luckily, some problems are easier in practice. Three-edge-colouring cubic toroidal graphs appears to be one of those things. To find colourings, we use a Boolean Satisfiability Solver or SAT solver. A SAT problem is a set of statements about a set of boolean variables \([x_1, x_2\ldots]\), such as “\(x_1\) or not \(x_3\) is true”. A solution to a SAT problem is a assignment \(x_i \in {0,1}\) that satisfies all the statements  [15]. General purpose, high performance programs for solving SAT problems have been an area of active research for decades  [16]. Such programs are useful because, by the Cook-Levin theorem  [17,18], any NP problem can be encoded (in polynomial time) as an instance of a SAT problem. This property is what makes SAT one of the subset of NP problems called NP-Complete. It is a relatively standard technique in the computer science community to solve NP problems by first transforming them to SAT instances and then using an off-the-shelf SAT solver. The output of this can then be mapped back to the original problem domain.

    +

    Whether graph colouring problems are in NP or P seems to depend delicately on the class of graphs considered, the maximum degree and the number of colours used. It is therefore possible that a polynomial time algorithm may exist for our problem. However using a SAT solver turns out to be fast enough in practice that it is by no means the rate limiting step for generating and solving instances of the AK model. In appendix A.3 I detail the specifics of how I mapped edge-colouring problems to SAT instances and show a breakdown of where the computational effort is spent, the majority being on diagonalisation.

    Mapping between flux sectors and bond sectors

    @@ -106,11 +105,11 @@ image:

    In the AK model, going from the bond sector to flux sector is done simply from the definition of the fluxes

    \[ \phi_i = \prod_{(j,k) \; \in \; \partial \phi_i} i u_{jk}.\]

    -

    The reverse, constructing the bond sector \(\{u_{jk}\}\) that corresponds to a particular flux sector \(\{\{\Phi_i\}\) is not so trivial. The algorithm, shown visually in fig. 3 is this:

    +

    The reverse, constructing a bond sector \(\{u_{jk}\}\) that corresponds to a particular flux sector \(\{\{\Phi_i\}\) is not so trivial. The algorithm I used, shown visually in fig. 3 is this:

    1. Fix the gauge by choosing some arbitrary \(u_{jk}\) configuration. In practice, we use \(u_{jk} = +1\). This chooses an arbitrary one of the four topological sectors.

    2. Compute the current flux configuration and how it differs from the target one. Consider any plaquette that differs from the target as a defect.

      3. -

    3. Find any adjacent pairs of defects and flip the \(u_jk\) between them. This leaves a set of isolated defects.

      4. +

    4. Find any adjacent pairs of defects and flip the \(u_{jk}\) between them. This leaves a set of isolated defects.

    5. Pair the defects up using a greedy algorithm and compute paths along the dual lattice between each pair of plaquettes using A*. Flipping the corresponding set of bonds transports one flux to the other and annihilates both.

    Next Section: Results

    diff --git a/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html b/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html index aa304ba..d978f4f 100644 --- a/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html +++ b/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html @@ -44,12 +44,16 @@ image:
  • The Ground State Flux Sector
  • Ground State Phase Diagram
  • Anderson Transition to a Thermal Metal
    • -
  • Conclusion
    • +
  • Discussion and Conclusion +
  • Bibliography
    • @@ -68,12 +72,16 @@ image:
  • The Ground State Flux Sector
  • Ground State Phase Diagram
  • Anderson Transition to a Thermal Metal
    • -
  • Conclusion
    • +
  • Discussion and Conclusion +
  • Bibliography
    • @@ -86,28 +94,28 @@ image:

    Results

    +

    This section contains our results on the AK model, we first look at how we checked numerically that Lieb’s theorem generalises to our model. Next we compute the ground state diagram and look at the two phases that arise there. We then use a local Chern marker and the presence of edge modes to characterise these phases as having Abelian or non-Abelian statistics. Finally we look at the finite temperature behaviour of the model.

    The Ground State Flux Sector

    -

    Here I will discuss the numerical evidence that our guess for the ground state flux sector is correct. We will do this by enumerating all the flux sectors of many separate system realisations. However we have two seemingly irreconcilable problems. Finite size effects have a large energetic contribution for small systems  [1] so we would like to perform our analysis for very large lattices. However for an amorphous system with \(N\) plaquettes, \(2N\) edges and \(3N\) vertices we have \(2^{N-1}\) flux sectors to check and diagonalisation scales with \(\mathcal{O}(N^3)\). That exponential scaling makes it infeasible to work with lattices much larger than \(16\) plaquettes.

    -

    To get around this we instead look at periodic systems with amorphous unit cells. For a similarly sized periodic system with \(A\) unit cells and \(B\) plaquettes in each unit cell where \(N \sim AB\) things get much better. We can use Bloch’s theorem to diagonalise this system in about \(\mathcal{0}(A B^3)\) operations, and more importantly there are only \(2^{B-1}\) flux sectors to check. We fully enumerated the flux sectors of ~25,000 periodic systems with disordered unit cells of up to \(B = 16\) plaquettes and \(A = 100\) unit cells. However, showing that our guess is correct for periodic systems with disordered unit cells is not quite convincing on its own as we have effectively removed longer-range disorder from our lattices.

    -

    The second part of the argument is to show that the energetic effect of introducing periodicity scales away as we go to larger system sizes and has already diminished to a small enough value at 16 plaquettes, which is indeed what we find. From this we argue that the results for small periodic systems generalise to large amorphous systems. In the isotropic case (\(J^\alpha = 1\)), our conjecture correctly predicted the ground state flux sector for all of the lattices we tested. For the toric code phase (\(J^x = J^y = 0.25, J^z = 1\)) all but around (\(\sim 0.5 \%\)) lattices had ground states conforming to our conjecture. In these cases, the energy difference between the true ground state and our prediction was on the order of \(10^{-6} J\). It is unclear whether this is a finite size effect or something else.

    -

    The spin Kitaev Hamiltonian is real and therefore has time reversal symmetry (TRS). However in the ground state the flux \(\phi_p\) through any plaquette with an odd number of sides has imaginary eigenvalues \(\pm i\). Thus, states with a fixed flux sector spontaneously break time reversal symmetry. This was first described by Yao and Kivelson for a translation invariant Kitaev model with odd sided plaquettes  [2].

    +

    We will check that Lieb’s theorem generalises to our model by enumerating all the flux sectors of many separate amorphous lattice realisations. However we have two seemingly irreconcilable problems. Finite size effects have a large energetic contribution for small systems  [1] so we would like to perform our analysis for very large lattices. However for an amorphous system with \(N\) plaquettes, \(2N\) edges and \(3N\) vertices we have \(2^{N-1}\) flux sectors to check and diagonalisation scales with \(\mathcal{O}(N^3)\). That exponential scaling makes it difficult to work with lattices much larger than \(16\) plaquettes with the resources.

    +

    To get around this we instead look at periodic systems with amorphous unit cells. For a similarly sized periodic system with \(A\) unit cells and \(B\) plaquettes in each unit cell where \(N \sim AB\) things get much better. We can use Bloch’s theorem to diagonalise this system in about \(\mathcal{O}(A B^3)\) operations, and more importantly there are only \(2^{B-1}\) flux sectors to check. We fully enumerated the flux sectors of \(\sim\) 25,000 periodic systems with disordered unit cells of up to \(B = 16\) plaquettes and \(A = 100\) unit cells. However, showing that our guess is correct for periodic systems with disordered unit cells is not quite convincing on its own as we have effectively removed longer-range disorder from our lattices.

    +

    The second part of the argument is to show that the energetic effect of introducing periodicity scales away as we go to larger system sizes and has already diminished to a small enough value at 16 plaquettes, which is indeed what we find. From this we argue that the results for small periodic systems generalise to large amorphous systems. In the isotropic case (\(J^\alpha = 1\)), Lieb’s theorem correctly predicts the ground state flux sector for all of the lattices we tested. For the toric code phase (\(J^x = J^y = 0.25, J^z = 1\)) all but around (\(\sim 0.5 \%\)) lattices had ground states conforming to our conjecture. In these cases, the energy difference between the true ground state and our prediction was on the order of \(10^{-6} J\).

    +

    The spin Kitaev Hamiltonian is real and therefore has time reversal symmetry. However in the ground state the flux \(\phi_p\) through any plaquette with an odd number of sides has imaginary eigenvalues \(\pm i\). Thus, states with a fixed flux sector spontaneously break time reversal symmetry. Kiteav noted this in his original paper but it was first explored in a concrete model by Yao and Kivelson for a translation invariant Kitaev model with odd sided plaquettes  [2].

    So we have flux sectors that come in degenerate pairs, where time reversal is equivalent to inverting the flux through every odd plaquette, a general feature for lattices with odd plaquettes  [3,4]. This spontaneously broken symmetry serves a role analogous to the external magnetic field in the original honeycomb model, leading the AK model to have a non-Abelian anyonic phase without an external magnetic field.

    Ground State Phase Diagram

    -

    The triangular phase \(J_x, J_y, J_z\) phase diagram of this family of models arises from setting the energy scale with \(J_x + J_y + J_z = 1\), the intersection of this plane and the unit cube is what yields the equilateral triangles seen in diagrams like fig. 1. The KH model has an Abelian, gapped phase in the anisotropic region (the A phase) and is gapless in the isotropic region. The introduction of a magnetic field breaks the chiral symmetry, leading to the isotropic region becoming a gapped, non-Abelian phase, the B phase.

    -

    Similar to the Kitaev Honeycomb model with a magnetic field, we find that the amorphous model is only gapless along critical lines, see fig. 1 (Left). Interestingly, in finite size systems the gap closing exists in only one of the four topological sectors though the sectors must become degenerate in the thermodynamic limit. Nevertheless this could be a useful way to define the (0, 0) topological flux sector for the amorphous model which otherwise has no natural way to choose it.

    -

    In the honeycomb model, the phase boundaries are located on the straight lines \(|J^x| = |J^y| + |J^x|\) and permutations of \(x,y,z\). These are shown as dotted lines on ~fig. 1 (Right). We find that on the amorphous lattice these boundaries exhibit an inward curvature, similar to honeycomb Kitaev models with flux or bond disorder  [5–10].

    +

    The triangular \(J_x, J_y, J_z\) phase diagram of this family of models arises from setting the energy scale with \(J_x + J_y + J_z = 1\). The intersection of this plane and the unit cube is what yields the equilateral triangles seen in diagrams like fig. 1. The KH model has an Abelian, gapped phase in the anisotropic region (the A phase) and is gapless in the isotropic region. The introduction of a magnetic field breaks the chiral symmetry, leading to the isotropic region becoming a gapped, non-Abelian phase, the B phase.

    +

    Similar to the KH model with a magnetic field, we find that the amorphous model is only gapless along critical lines, see fig. 1 (Left). Interestingly, in finite size systems the gap closing exists in only one of the four topological sectors though the sectors become degenerate in the thermodynamic limit. Nevertheless this could be a useful way to define the (0, 0) topological flux sector for the amorphous model which otherwise has no natural way to choose it.

    +

    In the honeycomb model, the phase boundaries are located on the straight lines \(|J^x| = |J^y| \;+ \;|J^x|\) and permutations of \(x,y,z\). These are shown as dotted lines in fig. 1 (Right). We find that on the amorphous lattice these boundaries exhibit an inward curvature, similar to honeycomb Kitaev models with flux or bond disorder  [5–10].

    Figure 1: The phase diagram of the model can be characterised by an equilateral triangle whose corners indicate points where J_\alpha = 1, J_\beta = J_\gamma = 0 while the centre denotes J_x = J_y = J_z. (Center) To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners of the phase diagram. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them (in green) has a true gap closing. (Right) The Abelian A_\alpha phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the Chern number \nu changes from 0 to \pm 1 from the A phases to the B phase. Indeed the gap must close in order for the Chern number to change  [11].
    -
    -

    Abelian or non-Abelian of the Gapped Phase

    -

    The two phases of the amorphous model are clearly gapped, though later I’ll double check this with finite size scaling.

    -

    The next question is: do these phases support excitations with trivial, Abelian or non-Abelian statistics? To answer that we turn to Chern numbers  [12–14]. As discussed earlier the Chern number is a quantity intimately linked to both the topological properties and the anyonic statistics of a model. Here we will make use of the fact that the Abelian/non-Abelian character of a model is linked to its Chern number  [1]. The Chern number is only defined for the translation invariant case because it relies on integrals defined in k-space. We instead use a family of real space generalisations of the Chern number that work for amorphous systems exist called local topological markers  [15–17], indeed Kitaev defines one in his original paper on the KH model  [1].

    -

    Here we use the crosshair marker of  [18] because it works well on smaller systems. We calculate the projector \(P = \sum_i |\psi_i\rangle \langle \psi_i|\) onto the occupied fermion eigenstates of the system in open boundary conditions. The projector encodes local information about the occupied eigenstates of the system and in gapped systems it is exponentially localised  [19]. The name crosshair comes from the fact that the marker is defined with respect to a particular point \((x_0, y_0)\) by step functions in x and y

    +
    +

    Abelian or non-Abelian statistics of the Gapped Phase

    +

    The two phases of the amorphous model are gapped as we can see from the finite size scaling of fig. 4. The next question is: do these phases support excitations with trivial, Abelian or non-Abelian statistics? To answer that we turn to Chern numbers  [12–14]. As discussed earlier the Chern number is a quantity intimately linked to both the topological properties and the anyonic statistics of a model. Here we will make use of the fact that the Abelian/non-Abelian character of a model is linked to its Chern number  [1]. The Chern number is only defined for the translation invariant case so we instead use a family of real space generalisations of the Chern number that work for amorphous systems called local topological markers  [15–17].

    +

    There are many possible choices here, indeed Kitaev defines one in his original paper on the KH model  [1]. Here we use the crosshair marker of  [18] because it works well on smaller systems. We calculate the projector \(P = \sum_i |\psi_i\rangle \langle \psi_i|\) onto the occupied fermion eigenstates of the system in open boundary conditions. The projector encodes local information about the occupied eigenstates of the system and in gapped systems it is exponentially localised  [19]. The name crosshair comes from the fact that the marker is defined with respect to a particular point \((x_0, y_0)\) by step functions in x and y

    \[\begin{aligned} \nu (x, y) = 4\pi \; \Im\; \mathrm{Tr}_{\mathrm{B}} \left ( @@ -115,16 +123,16 @@ image: \right ), \end{aligned}\]

    when the trace is taken over a region \(B\) around \((x_0, y_0)\) that is large enough to include local information about the system but does not come too close to the edges. If these conditions are met then then this quantity will be very close to quantised to the Chern number, see fig. 2. We’ll use the crosshair marker to assess the Abelian/non-Abelian character of the phases.

    -

    In the A phase of the amorphous model we find that \(\nu=0\) and hence the excitations have Abelian character, similar to the honeycomb model. This phase is thus the amorphous analogue of the Abelian toric-code quantum spin liquid  [20]. The B phase has \(\nu=\pm1\) so is a non-Abelian chiral spin liquid (CSL) similar to that of the Yao-Kivelson model  [3]. The CSL state is the the magnetic analogue of the fractional quantum Hall state  [21]. Hereafter we focus our attention on this phase.

    +

    In the A phase of the amorphous model we find that \(\nu=0\) and hence the excitations have Abelian character, similar to the honeycomb model. This phase is thus the amorphous analogue of the Abelian toric-code QSL  [20]. The B phase has \(\nu=\pm1\) so is a non-Abelian Chiral Spin Liquid (CSL) similar to that of the Yao-Kivelson model  [3]. The CSL state is the magnetic analogue of the fractional quantum Hall state  [21]. Hereafter we focus our attention on this phase.

    -Figure 2: (Center) The crosshair marker  [18], a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian. - +Figure 2: (Center) The crosshair marker  [18], a local topological marker, evaluated on the Amorphous Kitaev model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian. +

    Edge Modes

    Chiral Spin Liquids support topological protected edge modes on open boundary conditions  [22]. Fig. 3 shows the probability density of one such edge mode. It is near zero energy and exponentially localised to the boundary of the system. While the model is gapped in periodic boundary conditions (i.e on the torus) these edge modes appear in the gap when the boundary is cut.

    -

    The localization of the edge modes can be quantified by their inverse participation ratio (IPR), \[\mathrm{IPR} = \int d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau},\] where \(L\sim\sqrt{N}\) is the linear dimension of the amorphous lattices and \(\tau\) the dimensional scaling exponent of IPR. This is relevant because localised in-gap states do not participate in transport and hence do not turn band insulators into metals. It is only when the gap fills with extended states that we get a metallic state.

    +

    The localisation of the edge modes can be quantified by their inverse participation ratio (IPR) and its scaling with system size \(\tau\), \[\mathrm{IPR} = \int d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau},\] where \(L\sim\sqrt{N}\) is the linear dimension of the amorphous lattices. This is relevant because localised in-gap states do not participate in transport and hence do not turn band insulators into conductive metals. It is only when the gap fills with extended states that we get a conductive state.

    Figure 3: (a) The density of one of the topologically protected edge states in the B phase. (Below) the log density plotted along the black path showing that the state is exponentially localised. (a)/(b) The density of states of the corresponding lattice in (a) periodic boundary conditions, (b) open boundary conditions. The colour of the bars shows the mean log IPR for each energy window. Cutting the boundary fills the gap with localised states. @@ -133,45 +141,45 @@ image:

    Anderson Transition to a Thermal Metal

    -

    Previous work on the honeycomb model at finite temperature has shown that the B phase undergoes a thermal transition from a quantum spin liquid phase a to a thermal metal phase  [23].

    -

    This happens because at finite temperature, thermal fluctuations lead to spontaneous vortex-pair formation. As discussed previously these fluxes are dressed by Majorana bounds states and the composite object is an Ising-type non-Abelian anyon  [24]. The interactions between these anyons are oscillatory similar to the RKKY exchange and decay exponentially with separation  [25–27]. At sufficient density, the anyons hybridise to a macroscopically degenerate state known as thermal metal  [25]. At close range the oscillatory behaviour of the interactions can be modelled by a random sign which forms the basis for a random matrix theory description of the thermal metal state.

    -

    The amorphous chiral spin liquid undergoes the same form of Anderson transition to a thermal metal state. Markov Chain Monte Carlo would be necessary to simulate this in full detail  [23] but in order to avoid that complexity in the current work we instead opted to use vortex density \(\rho\) as a proxy for temperature. We give each plaquette probability \(\rho\) of being a vortex, possibly with one additional adjustment to preserve overall vortex parity. This approximation is exact in the limits \(T = 0\) (corresponding to \(\rho = 0\)) and \(T \to \infty\) (corresponding to \(\rho = 0.5\)) while at intermediate temperatures there may be vortex-vortex correlations that are not captured by positioning vortices using uncorrelated random variables.

    -

    First we performed a finite size scaling to that the presence of a gap in the CSL ground state and absence of a gap in the thermal phase are both robust as we go to larger systems, see fig. 4.

    Figure 4: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03
    -

    Next we evaluated the fermionic density of states (DOS), Inverse Participation Ratio (IPR) and IPR scaling exponent \(\tau\) as functions of the vortex density \(\rho\), see fig. 5. This leads to a nice picture of what happens as we raise the temperature of the system away from the gapped, insulating CSL phase. At small \(\rho\), states begin to populate the gap but they have \(\tau\approx0\), indicating that they are localised states pinned to the vortices, and the system remains insulating. At large \(\rho\), the in-gap states merge with the bulk band and become extensive, closing the gap, and the system transitions to the thermal metal phase.

    +

    Previous work on the honeycomb model, at finite temperature has shown that the B phase undergoes a thermal transition from a QSL phase to a thermal metal phase  [23]. This happens because at finite temperature, thermal fluctuations lead to spontaneous vortex-pair formation. As discussed previously, these fluxes are dressed by Majorana bounds states and the composite object is an Ising-type non-Abelian anyon  [24]. The interactions between these anyons are oscillatory, similar to the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange and decay exponentially with separation  [25–27]. At sufficient density, the anyons hybridise to a macroscopically degenerate state known as thermal metal  [25]. At close range the oscillatory behaviour of the interactions can be modelled by a random sign which forms the basis for a random matrix theory description of the thermal metal state.

    +

    The amorphous chiral spin liquid undergoes the same form of Anderson transition to a thermal metal state. Markov Chain Monte Carlo would be necessary to simulate this in full detail  [23] but in order to avoid that complexity in the current work we instead opted to use vortex density \(\rho\) as a proxy for temperature. We give each plaquette the probability \(\rho\) of being a vortex, possibly with one additional adjustment to preserve overall vortex parity. This approximation is exact in the limits \(T = 0\) (corresponding to \(\rho = 0\)) and \(T \to \infty\) (corresponding to \(\rho = 0.5\)) while at intermediate temperatures there may be vortex-vortex correlations that are not captured by our uncorrelated vortex placement.

    +

    First we performed a finite size scaling to show that the presence of a gap in the CSL ground state and absence of a gap in the thermal metal phase are both robust as we go to larger systems, see fig. 4.

    +

    Next we evaluated the fermionic density of states (DOS), Inverse Participation Ratio and IPR scaling exponent \(\tau\) as functions of the vortex density \(\rho\), see fig. 5. This leads to a nice picture of what happens as we raise the temperature of the system away from the gapped, insulating CSL phase. At small \(\rho\), states begin to populate the gap but they have \(\tau\approx0\), indicating that they are localised states pinned to the vortices, and the system remains insulating. At large \(\rho\), the in-gap states merge with the bulk band and become extensive, closing the gap, and the system transitions to the thermal metal phase.

    Figure 5: (Top) Density of states and (Bottom) scaling exponent \tau of the amorphous Kitaev model as a vortex density \rho is increased. The scaling exponent \tau is the exponent with which the inverse participation ratio scales with system size. It gives a measure of the degree of localisation of the states in each (E/J, \rho) bin. At zero \rho we have the gapped ground state. At small \rho, states begin to populate the gap. These states have \tau\approx0, indicating that they are localised states pinned to fluxes, and the system remains insulating. As \rho increases further, the in-gap states merge with the bulk band and become extensive, fully closing the gap, and the system transitions to a thermal metal phase.
    -

    The thermal metal phase has a signature logarithmic divergence at zero energy and oscillations in the DOS. These signatures can be shown to occur by a recursive argument that involves mapping the original model onto a Majorana model with interactions that take random signs which can itself be mapped onto a coarser lattice with lower energy excitations and so on. This can be repeating indefinitely, showing the model must have excitations at arbitrarily low energies in the thermodynamic limit  [23,28].

    -

    These signatures for our model and for the honeycomb model are shown in fig. 6. They do not occur in the honeycomb model unless the chiral symmetry is broken by a magnetic field.

    +

    The thermal metal phase has a signature logarithmic divergence at zero energy and oscillations in the DOS. These signatures can be shown to occur by a recursive argument that involves mapping the original model onto a Majorana model with interactions that take random signs which can itself be mapped onto a coarser lattice with lower energy excitations and so on. This can be repeating indefinitely, showing the model must have excitations at arbitrarily low energies in the thermodynamic limit  [23,28]. These signatures are shown in fig. 6 for our model and for the KH model. They do not occur in the KH model unless the chiral symmetry is broken by a magnetic field.

    Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [23,28]. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit.
    -

    We found a small number of lattices for which the ground state conjecture did not correctly predict the true ground state flux sector. I see two possibilities for what could cause this. Firstly it could be a a finite size effect that is amplified by certain rare lattice configurations. It would be interesting to try to elucidate what lattice features are present when the ground state conjecture fails. Alternatively, it might be telling that the ground state conjecture failed in the toric code A phase where the couplings are anisotropic. We showed that the colouring does not matter in the B phase. However an avenue that I did not explore was whether the particular choice of colouring for a lattice affects the physical properties in the toric code A phase. It is possible that some property of the particular colouring chosen is what leads to failure of the ground state conjecture here.

    -

    Conclusion

    -

    In this chapter we have looked at an extension of the Kitaev honeycomb model to amorphous lattices with coordination number three. We discussed a method to construct arbitrary trivalent lattices using Voronoi partitions, how to embed them onto the torus and how to edge-colour them using a SAT solver. We showed numerically that the ground state flux sector of the model is given by a simple extension of Lieb’s theorem. The model has two gapped quantum spin liquid phases. The two phases support excitations with different anyonic statistics, Abelian and non-Abelian, distinguished using a real-space generalisation of the Chern number  [18]. The presence of odd-sided plaquettes in the model results in spontaneous breaking of time reversal symmetry, leading to the emergence of a chiral spin liquid phase. Finally we showed evidence that the amorphous system undergoes an Anderson transition to a thermal metal phase, driven by the proliferation of vortices with increasing temperature.

    -

    The AK model is an exactly solvable model of the chiral QSL state, one of the first models to exhibit a topologically non-trivial quantum many-body phase on an amorphous lattice. As such this study provides a number of future lines of research.

    -

    Experimental Realisations and Signatures

    -

    We should also consider whether a physical amorphous system that supports a QSL ground state could exist. The search for translation invariant Kitaev systems is already motivated by the prospect of a physically realised QSL state, Majorana fermions and direct access to a system with emergent \(\mathbb{Z}_2\) gauge physics  [29]. An amorphous Kitaev model would provide all this but in addition the possibility of exploring the CSL state as well as potentially very different routes to a physical realisation. One route would be to ask if any crystalline Kitaev material candidates can be heated and rapidly quenched  [30–32] to produce amorphous analogues that might preserve enough of their local structure to support a QSL state.

    -

    Instead looking to more designer materials, metal organic frameworks (MOFs) could present a platform for a synthetic Kitaev material. These materials are composed of repeating units of large organic molecular frameworks coordinated with metal ions. Amorphous MOFs can be generated with mechanical processes that introduce disorder into crystalline MOFs  [33]. There have been recent proposals for realizing strong Kitaev interactions  [34] in them as potential signatures of a resonating valence bond QSL state in MOFs with Kagome geometry  [35]. Finally MOFs are composed of large synthetic molecules so may provide more opportunity for fine tuning to target particular physics than than elemental compounds. There have also been proposals to realise Kitaev physics in optical lattice experiments  [36,37] which can also support amorphous lattices  [38].

    -

    A physical realisation in either an amorphous compound or a MOF would likely entail a high degree of defects. Amorphous silicon for instance tends to contain a high degree of dangling bonds which must be passivated by hydrogenation to improve its physical properties  [39]. In both cases, if we assume that Kitaev physics can be realised by crystalline systems, it is not clear if the necessary superexchange couplings would survive the addition of disorder to the MOF lattice. It would therefore make sense to examine how robust the CSL ground state of the AK model is to additional disorder in the Hamiltonian, for example mis-colourings of the bonds, vertex degree disorder and disorder in coupling strengths. Relatedly, one could look at perturbations to the Hamiltonian that break integrability  [40–44].

    -

    Considering experimental signatures, we expect that the chiral amorphous QSL will display a half-quantized thermal Hall effect similar to the magnetic field induced behaviour of KH materials  [45–48]. Alternatively, the CSL state could be characterized by local probes such as spin-polarized scanning tunnelling microscopy  [49–51] and the thermal metal phase displays characteristic longitudinal heat transport signatures  [24].

    -

    Local perturbations such as those that might come from an atomic force microscope could potentially be used to create and control vortices  [52] To this end it may make sense to look at how the move to amorphous lattices affects vortex time dynamics in perturbed KH models  [53].

    -

    Given the lack of unambiguous signatures of the QSL state it can be hard to distinguish the effects of the QSL state from the effect of disorder. So introducing topological disorder from amorphous lattices may pose considerable experimental challenges. Three dimensional realisations could get around this as they would be expected to have a true FTPT to the thermal metal state that could be a useful experimental signature  [54,55]. Three dimensional Kitaev systems can also support CSL ground states  [56].

    -

    Thermodynamics

    -

    The KH model can be extended to three dimensional tri-coordinate lattices  [54,55] or it can be generalised to an exactly solvable spin-\(\tfrac{3}{2}\) model on four-coordinate lattices  [2,57–68]. In  [57] the two dimensional square lattice with 4 bond types (\(J_w, J_x, J_y, J_z\)) is considered. Since Voronoi partitions in three dimensions produce lattices of degree four, one interesting generalisation of this work would be to look at the spin-\(\tfrac{3}{2}\) Kitaev model on amorphous lattices.

    -

    We did not perform a full MCMC simulation of the AK model at finite temperature but the possible extension of the model to three dimensions with an FTPT would motive this full analysis in different dimensions. This would be a numerically challenging task but poses no conceptual barriers  [23,25,27]. Doing this would, firstly, allow one to look for possible violations of the Harris criterion  [69] for the Ising transition of the flux sector. Recall that topological disorder in two dimensions has radically different properties to that of other kinds of disorder due to the constraints imposed by the Euler equation and maintaining coordination number which allows it to violate otherwise quite general rules like the Harris criterion  [70,71]. Second, incorporating the projector in addition to MCMC would allow for a full investigation of whether the effect of topological degeneracy is apparent at finite temperatures as is done in  [23].

    -

    Next, one could investigate whether a QSL phase may exist for for other models defined on amorphous lattices with a view to more realistic prospects of observation. For instance, it would be interesting to see if the properties of the Kitaev-Heisenberg model generalise from the honeycomb to the amorphous case [ [41];  [43];  [72];  [73];  [74];]. Alternatively we might look at other lattice construction techniques. For instance we could construct lattices by linking close points  [75] or create simplices from random sites  [76]. Lattices constructed these ways would like have a large number of lattice defects \(z \neq 3\) in the bulk, leading to persistent zero modes.

    -

    Overall, there has been surprisingly little research on amorphous quantum many body phases despite there being plenty of material candidates. I expect the exact chiral amorphous spin liquid to find many generalisation to realistic amorphous quantum magnets.

    -

    Next Chapter: 5 Conclusion

    +

    Discussion and Conclusion

    +

    In this chapter we have looked at an extension of the KH model to amorphous lattices with coordination number three. We discussed a method to construct arbitrary trivalent lattices using Voronoi partitions, how to embed them onto the torus and how to edge-colour them using a SAT solver. We showed numerically that the ground state flux sector of the model is given by a simple extension of Lieb’s theorem. The model has two gapped QSL phases. The two phases support excitations with different anyonic statistics, Abelian and non-Abelian, distinguished using a real-space generalisation of the Chern number  [18]. The presence of odd-sided plaquettes in the model resulted in spontaneous breaking of time reversal symmetry, leading to the emergence of a chiral spin liquid phase. Finally we showed evidence that the amorphous system undergoes an Anderson transition to a thermal metal phase, driven by the proliferation of vortices with increasing temperature. The AK model is an exactly solvable model of the chiral QSL state, one of the first models to exhibit a topologically non-trivial quantum many-body phase on an amorphous lattice. As such this study provides a number of future lines of research.

    +
    +

    Experimental Realisations and Signatures

    +

    We should consider whether a physical amorphous system that supports a QSL ground state could exist. The search for translation invariant Kitaev systems is already motivated by the prospect of a physically realised QSL state, Majorana fermions and direct access to a system with emergent \(\mathbb{Z}_2\) gauge physics  [29]. An amorphous Kitaev model would provide all this and in addition the possibility of exploring the CSL state and potentially very different routes to a physical realisation. One route would be to ask if any crystalline Kitaev material candidates can be heated and rapidly quenched  [30–32] to produce amorphous analogues that might preserve enough of their local structure to support a QSL state.

    +

    Considering more designer materials, metal organic frameworks (MOFs) could present a platform for a synthetic Kitaev material. These materials are composed of repeating units of large organic molecules coordinated with metal ions. Amorphous MOFs can be generated with mechanical processes that introduce disorder into crystalline MOFs  [33] and there have been recent proposals for realising strong Kitaev interactions  [34] in them as potential signatures of a resonating valence bond QSL state in MOFs with Kagome geometry  [35]. Finally, MOFs are composed of large synthetic molecules so may provide more opportunity for fine tuning to target particular physics than with ionic compounds. There have also been proposals to realise Kitaev physics in optical lattice experiments  [36,37] which can also support amorphous lattices  [38].

    +

    A physical realisation in either an amorphous compound or a MOF would likely entail a high degree of defects. Amorphous silicon, for instance, tends to contain dangling bonds which must be passivated by hydrogenation to improve its physical properties  [39]. In both cases, if we assume that Kitaev physics can be realised by crystalline systems, it is not clear if the necessary superexchange couplings would survive the addition of disorder to the lattice. It would therefore make sense theoretically to examine how robust the CSL ground state of the AK model is to additional disorder in the Hamiltonian, for example mis-colourings of the bonds, vertex degree disorder and disorder in coupling strengths. Relatedly, one could look at perturbations to the Hamiltonian that break integrability  [40–44].

    +

    Considering experimental signatures, we expect that the chiral amorphous QSL will display a half-quantised thermal Hall effect similar to the magnetic field induced behaviour of KH materials  [45–48]. Alternatively, the CSL state could be characterised by local probes such as spin-polarised scanning tunnelling microscopy  [49–51] while the thermal metal phase displays characteristic longitudinal heat transport signatures  [24]. Local perturbations, such as those that might come from an atomic force microscope, could potentially be used to create and control vortices  [52]. To this end it one could look at how the move to amorphous lattices affects vortex time dynamics in perturbed KH models  [53].

    +

    Given the lack of unambiguous signatures of the QSL state, it can be hard to distinguish the effects of the QSL state from the effect of disorder. So introducing topological disorder may only increase the experimental challenges. Three dimensional realisations could get around this as they would be expected to have a true Finite-Temperature Phase Transition (FTPT) to the thermal metal state that could be a useful experimental signature  [54,55]. Three dimensional Kitaev systems can also support CSL ground states  [56].

    +
    +
    +

    Thermodynamics

    +

    The KH model can be extended to 3D either on trivalent lattices  [54,55] or it can be generalised to an exactly solvable spin-\(\tfrac{3}{2}\) model on 3D four-coordinate lattices  [2,57–68]. In  [57], the 2D square lattice with 4 bond types (\(J_w, J_x, J_y, J_z\)) is considered. Since Voronoi partitions in 3D produce lattices of degree four, one interesting generalisation of this work would be to look at the spin-\(\tfrac{3}{2}\) Kitaev model on amorphous lattices.

    +

    We did not perform a full Markov Chain Monte Carlo (MCMC) simulation of the AK model at finite temperature but the possible extension to a 3D model with an FTPT would motivate this full analysis. This MCMC simulation would be a numerically challenging task but poses no conceptual barriers  [23,25,27]. Doing this would, first, allow one to look for possible violations of the Harris criterion  [69] for the Ising transition of the flux sector. Recall that topological disorder in 2D has radically different properties to that of other kinds of disorder due to the constraints imposed by the Euler equation and maintaining coordination number which allows it to violate otherwise quite general rules like the Harris criterion  [70,71]. Second, incorporating the projector in addition to MCMC would allow for a full investigation of whether the effect of topological degeneracy is apparent at finite temperatures, this is done for the KH model in  [23].

    +

    Next, one could investigate whether a QSL phase may exist for other models defined on amorphous lattices with a view to more realistic prospects of observation. Do the properties of the Kitaev-Heisenberg model generalise from the honeycomb to the amorphous case?  [41,43,72–74] Alternatively we might look at other lattice construction techniques. For instance we could construct lattices by linking close points  [75] or create simplices from random sites  [76]. Lattices constructed using these methods would likely have a large number of lattice defects where \(z \neq 3\) in the bulk, leading to many localised Majorana zero modes.

    +

    We found a small number of lattices for which Lieb’s theorem did not correctly predict the true ground state flux sector. I see two possibilities for what could cause this. Firstly it could be a a finite size effect that is amplified by certain rare lattice configurations. It would be interesting to try to elucidate what lattice features are present when Lieb’s theorem fails. Alternatively, it might be telling that the ground state conjecture failed in the toric code A phase where the couplings are anisotropic. We showed that the colouring does not matter in the B phase. However an avenue that I did not explore was whether the particular choice of colouring for a lattice affects the physical properties in the toric code A phase. It is possible that some property of the particular colouring chosen is what leads to these rare failures of Lieb’s theorem.

    +

    Overall, there has been surprisingly little research on amorphous quantum many-body phases despite there being plenty of material candidates. I expect the exact chiral amorphous spin liquid to find many generalisations to realistic amorphous quantum magnets.

    +

    Next Chapter: 5 Conclusion

    +

    Bibliography

    diff --git a/_thesis/5_Conclusion/5_Conclusion.html b/_thesis/5_Conclusion/5_Conclusion.html index e37bf94..01aef8e 100644 --- a/_thesis/5_Conclusion/5_Conclusion.html +++ b/_thesis/5_Conclusion/5_Conclusion.html @@ -38,14 +38,6 @@ image:
    @@ -59,65 +51,84 @@ image: -
    -

    5 Conclusion

    -
    -
    -
    -

    5 Conclusion

    -

    Using exactly solvable systems as a way to look at the physics of many-body interacting systems. Another theme from the two models is that longer range correlations from criticality in the LRFK model and anti-correlations in the topological disorder in the AK model, lead to a wider range of effects that short range correlations.

    -

    paragraph about topological order as new addition to the pantheon of spontaneously broken symmetries

    -

    FK model as a way to probe the Mott insulator state. Also the Mott insulator gives rise to the QSl and the doped Mott Insulator may be the source of the sought after High-\(T_c\) superconductor. The concept of quantum orders is relevant because for instance, if we can classify the kinds of order in the MI state, we can classify the kinds of high-\(T_c\) theories that can emerge from them.

    -

    Xiao-Gang Wen  [1] when talks about quantum orders as a those that arise within quantum states at zero temperature, included QSLs, FQH states and superconductors1. He also argues that the High-\(T_c\) superconductors is in terms of them being doped Mott insulators so that we should try to understand the QSL which emerges from the undoped Mott insulator (at half filling).

    -

    The existence of distinct, spatially limited quasiparticle excitations is not obvious.

    -

    emergent gauge physics, could condensed matter systems be useful in understanding the standard model too?

    -

    Electron-electron interactions play a dominant role in determining electronic and thermodynamic properties in these strongly correlated materials.

    -

    Specific examples where strongly correlated materials may figure prominently are high temperature superconductors and hard magnets without rare earth elements.

    -
    -
    -

    Material Realisations

    -
    -

    Amorphous Materials

    -
    -
    -

    Metal Organic Frameworks

    -
    -
    -
    -

    Discussion

    -
    -
    -

    Outlook

    -

    Next Chapter: Appendices

    -
    +

    This thesis has focussed on two strongly correlated systems. As is the case with many strongly correlated systems, their many-body ground states can be complex and often cannot be reduced to or even adiabatically connected to a product state. In this work, we looked at the Falicov-Kimball (FK) model and the Kitaev Honeycomb (KH) model and defined extensions to them: the Long-Range Falicov-Kimball (LRFK) and amorphous Kitaev (AK) models.

    +

    These models are all exactly solvable. They contain extensively many conserved charges which allow their Hamiltonians, and crucially, the interaction terms within them, to be written in quadratic form. This allows them to be solved using the theoretical machinery of non-interacting systems. In the case of the FK and LRFK models, this solvability arises from what is essentially a separation of timescales. The heavy particles move so slowly that they can be treated as stationary. In the KH and AK models, on the other hand, the origin of the conserved degrees of freedom is more complex. Here, the algebra of the Pauli matrices interacts with the trivalent lattices on which the models are defined, to give rise to an emergent \(\mathbb{Z}_2\) gauge field whose fluxes are conserved. This latter case is a beautiful example of emergence at play in condensed matter. The gauge and Majorana physics of the Kitaev models seem to arise spontaneously from nothing. Though, of course, this physics was hidden within the structure and local symmetries of the spin Hamiltonian all along.

    +

    At first glance, exactly solvable models can seem a little too fine tuned to be particularly relevant to the real world. Surely these models don’t spontaneously arise in nature? The models studied here provide two different ways to answer this. As we saw, the FK model arises quite naturally as a limit of the Hubbard model which is not exactly solvable. In fact, the FK model has been used as a way to understand more about the behaviour of the Hubbard model itself and of the Mott insulating state. We’ve also seen that it can provide insight into other phenomena such as disorder-free localisation. The KH model was not originally proposed as a model of any particular physical system. It was nevertheless a plausible microscopic Hamiltonian and, given its remarkable properties, it is little wonder that material candidates for Kitaev physics were quickly found. In neither case is the model expected to be a perfect description of any material, indeed more realistic corrections to each model are likely to break their integrability. Despite this, exactly solvable models, by virtue of being solvable, can provide important insights into the diverse physics of strongly correlated materials.

    +

    In chapter 3, we looked at a generalised FK model in 1D, the LRFK model. Metal-insulator transitions are a key theme of work on the FK model and our 1D extension to it was no exception. With the addition of long-range interactions, the model showed a similarly rich phase diagram as its higher dimensional cousins, allowing us to look at transitions between metallic states and, band, Anderson and Mott insulators. We also looked at thermodynamics in 1D and how thermal fluctuations of the conserved charges can lead to disorder-free localisation in the FK and LRFK models. Though the initial surprising results suggested the presence of a mobility edge in 1D it turned out to be a weak localisation effect present in finite sized systems. We propose that a topological variant of the LRFK model akin to the SSH model could be interesting as a point for further study. It could also be an interesting target for cold atom experiments that can naturally generate long-range interactions  [1].

    +

    That the Mott insulating state is a key part of the work on the LRFK model is fitting because Mott insulators are the main route to the formation of Quantum Spin Liquid (QSL) states which are themselves a primary driver of interest in the KH model and our extension, the AK model. In chapter 4, we addressed the question of whether frustrated magnetic interactions on amorphous lattices can give rise to quantum phases such as the QSL state and found that indeed they can. The AK model, a generalisation of the KH model to random lattices with fixed coordination number three, supports a kind of symmetry broken QSL state called a chiral spin liquid. We showed numerically that the ground state of the model follows a simple generalisation of Lieb’s theorem  [2–4]. As with other extensions of the KH model  [5], we found that removing the chiral symmetry of the lattice allows the model to support a gapped phase with non-Abelian anyon excitations. The broken lattice symmetry plays the role of the external magnetic field in the original KH model. Finally, like the KH model, finite temperature causes vortex defects to proliferate, causing a transition to a thermal metal state. We have discussed the prospect of whether AK model physics might be realisable in amorphous versions of known KH candidate materials  [6]. Alternatively, we might be able to engineer them in synthetic materials such as Metal Organic Frameworks where both Kitaev interactions and amorphous lattices have already been proposed  [7,8].

    +

    Unlike the 2D FK model and 1D LRFK models, the KH and AK models don’t have a Finite-Temperature Phase Transition (FTPT). They immediately disorder at any finite temperature  [4]. However, generalisations of the KH model to 3D do in general have an FTPT. Indeed, the role of dimensionality has been a key theme in this work. Both localisation and thermodynamic phenomena depend crucially on dimensionality, with thermodynamic order generally suppressed and localisation effects strengthened in low dimensions. The graph theory that underpins the KH and AK models itself also changes strongly with dimension. Voronisation in 2D produces trivalent lattices, on which the spin-\(1/2\) AK model is exactly solvable. Meanwhile in 3D, Voronisation gives us \(z=4\) lattices upon which a spin-\(3/2\) generalisation to the KH model is exactly solvable  [9–11]. Similarly, planar graphs are a uniquely 2D construct. Satisfying planarity imposes constraints on the connectivity of planar graphs leading amorphous planar graphs to have strong anti-correlations which can violate otherwise robust bounds like the Harris criterion  [12]. Contrast this with Anderson localisation in 1D where only longer range correlations in the disorder can produce surprising effects  [13–18].

    +

    Xiao-Gang Wen makes the case that one of the primary reasons to study QSLs is as a stepping stone to understanding the High-\(T_c\) superconductors  [19]. His logic is that since the High-\(T_c\) superconductors are believed to arise from doped Mott insulators, the QSLs, which arise from undoped Mott insulators, make a good jumping off point. This is where exactly solvable models like the FK and KH models shine. The FK model provides a tractable means to study superconductor like states in doped Mott insulators  [20], while the KH model gives us a tangible QSL state to play with. The extensions introduced in this work serve to explore how far we can push these models. Overall, I believe that the results presented here show that exactly solvable models can be a useful theoretical tool for understanding the behaviour of generic strongly correlated materials.

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    M. G. Yamada, H. Fujita, and M. Oshikawa, Designing Kitaev Spin Liquids in Metal-Organic Frameworks, Phys. Rev. Lett. 119, 057202 (2017).
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    F. M. Izrailev and A. A. Krokhin, Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder, Phys. Rev. Lett. 82, 4062 (1999).
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    F. M. Izrailev, A. A. Krokhin, and N. M. Makarov, Anomalous Localization in Low-Dimensional Systems with Correlated Disorder, Physics Reports 512, 125 (2012).
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    [1]
    X.-G. Wen, Quantum Orders and Symmetric Spin Liquids, Phys. Rev. B 65, 165113 (2002).
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    X.-G. Wen, Quantum Orders and Symmetric Spin Liquids, Phys. Rev. B 65, 165113 (2002).
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    P. Cai, W. Ruan, Y. Peng, C. Ye, X. Li, Z. Hao, X. Zhou, D.-H. Lee, and Y. Wang, Visualizing the Evolution from the Mott Insulator to a Charge-Ordered Insulator in Lightly Doped Cuprates, Nature Phys 12, 11 (2016).
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    1. Wen argues that superconductors cannot be characterised be a local order parameter in the way that superfluids can.↩︎

      2. -
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    diff --git a/_thesis/6_Appendices/A.3_Lattice_Generation.html b/_thesis/6_Appendices/A.3_Lattice_Generation.html index 57fcd14..3ceadbd 100644 --- a/_thesis/6_Appendices/A.3_Lattice_Generation.html +++ b/_thesis/6_Appendices/A.3_Lattice_Generation.html @@ -103,7 +103,7 @@ image:

    To require that exactly one of the variables be true, we can enforce that no pair of variables be true: -(r and b) -(r and g) -(b and g)

    However, these clauses are not in CNF form. Therefore, we also have to use the fact that -(a and b) = (-a OR -b). To enforce that at least one of these is true we simply OR them all together (r or b or g)

    To encode the fact that no adjacent edges can have the same colour, we emit a clause that, for each pair of adjacent edges, they cannot be both red, both green or both blue.

    -

    We get a solution or set of solutions from the solver, which we can map back to a labelling of the edges. fig. ¿fig:multiple_colourings? shows some examples.

    +

    We get a solution or set of solutions from the solver, which we can map back to a labelling of the edges.

    The solution presented here works well enough for our purposes. It does not take up a substantial fraction of the overall computation time, see +fig:times but other approaches could likely work.

    When translating problems to CNF form, there is often some flexibility. For instance, we used three boolean variables to encode the colour of each edge and, then, additional constraints to require that only one of these variables be true. An alternative method which we did not try would be to encode the label of each edge using two variables, yielding four states per edge, and then add a constraint that one of the states, say (true, true) is disallowed. This would, however, have added some complexity to the encoding of the constraint that no adjacent edges can have the same colour.

    The popular Networkx Python library uses a greedy graph colouring algorithm. It simply iterates over the vertices/edges/faces of a graph and assigns them a colour that is not already disallowed. This does not work for our purposes because it is not designed to look for a particular n-colouring. However, it does include the option of using a heuristic function that determine the order in which vertices will be coloured  [2,3]. Perhaps

    diff --git a/_thesis/6_Appendices/A.5_The_Projector.html b/_thesis/6_Appendices/A.5_The_Projector.html index 9674eac..ab8b0df 100644 --- a/_thesis/6_Appendices/A.5_The_Projector.html +++ b/_thesis/6_Appendices/A.5_The_Projector.html @@ -66,10 +66,10 @@ image:

    The Projector

    The projection from the extended space to the physical space will not be particularly important for the results presented here. However, the theory remains useful to explain why this is.

    -
    -Figure 1: The relationship between the different Hilbert spaces used in the solution. needs updating - -
    +

    The physical states are defined as those for which \(D_i |\phi\rangle = |\phi\rangle\) for all \(D_i\). Since \(D_i\) has eigenvalues \(\pm1\), the quantity \(\tfrac{(1+D_i)}{2}\) has eigenvalue \(1\) for physical states and \(0\) for extended states so is the local projector onto the physical subspace.

    Therefore, the global projector is \[ \mathcal{P} = \prod_{i=1}^{2N} \left( \frac{1 + D_i}{2}\right)\]

    for a toroidal trivalent lattice with \(N\) plaquettes \(2N\) vertices and \(3N\) edges. As discussed earlier, the product over \((1 + D_j)\) can also be thought of as the sum of all possible subsets \(\{i\}\) of the \(D_j\) operators, which is the set of all possible gauge symmetry operations.

    @@ -82,12 +82,12 @@ image:

    \[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i \prod_i^{2N} b^y_i \prod_i^{2N} b^z_i \prod_i^{2N} c_i\]

    The product over \(c_i\) operators reduces to a determinant of the Q matrix and the fermion parity, see  [1]. The only difference from the honeycomb case is that we cannot explicitly compute the factors \(p_x,p_y,p_z = \pm\;1\) that arise from reordering the b operators such that pairs of vertices linked by the corresponding bonds are adjacent.

    \[\prod_i^{2N} b^\alpha_i = p_\alpha \prod_{(i,j)}b^\alpha_i b^\alpha_j\]

    -

    However, they are simply the parity of the permutation from one ordering to the other and can be computed in linear time with a cycle decomposition cite.

    +

    However, they are simply the parity of the permutation from one ordering to the other and can be computed in linear time with a cycle decomposition  [2].

    We find that \[\mathcal{P}_0 = 1 + p_x\;p_y\;p_z\; \hat{\pi} \; \mathrm{det}(Q^u) \; \prod_{\{i,j\}} -iu_{ij}\]

    where \(p_x\;p_y\;p_z = \pm 1\) are lattice structure factors and \(\mathrm{det}(Q^u)\) is the determinant of the matrix mentioned earlier that maps \(c_i\) operators to normal mode operators \(b'_i, b''_i\). These depend only on the lattice structure.

    -

    \(\hat{\pi} = \prod{i}^{N} (1 - 2\hat{n}_i)\) is the parity of the particular many body state determined by fermionic occupation numbers \(n_i\). As discussed in  [1], \(\hat{\pi}\) is gauge invariant in the sense that \([\hat{\pi}, D_i] = 0\).

    -

    This implies that \(det(Q^u) \prod -i u_{ij}\) is also a gauge invariant quantity. In translation invariant models this quantity which can be related to the parity of the number of vortex pairs in the system  [2].

    -

    All these factors take values \(\pm 1\) so \(\mathcal{P}_0\) is 0 or 1 for a particular state. Since \(\mathcal{S}\) corresponds to symmetrising over all the gauge configurations and cannot be 0, once we have determined the single particle eigenstates of a bond sector, the true many body ground state has the same energy as either the empty state with \(n_i = 0\) or a state with a single fermion in the lowest level.

    +

    \(\hat{\pi} = \prod{i}^{N} (1 - 2\hat{n}_i)\) is the parity of the particular many-body state determined by fermionic occupation numbers \(n_i\). As discussed in  [1], \(\hat{\pi}\) is gauge invariant in the sense that \([\hat{\pi}, D_i] = 0\).

    +

    This implies that \(det(Q^u) \prod -i u_{ij}\) is also a gauge invariant quantity. In translation invariant models this quantity which can be related to the parity of the number of vortex pairs in the system  [3].

    +

    All these factors take values \(\pm 1\) so \(\mathcal{P}_0\) is 0 or 1 for a particular state. Since \(\mathcal{S}\) corresponds to symmetrising over all the gauge configurations and cannot be 0, once we have determined the single particle eigenstates of a bond sector, the true many-body ground state has the same energy as either the empty state with \(n_i = 0\) or a state with a single fermion in the lowest level.

    Bibliography

    @@ -95,8 +95,11 @@ image:
    [1]
    F. L. Pedrocchi, S. Chesi, and D. Loss, Physical solutions of the Kitaev honeycomb model, Phys. Rev. B 84, 165414 (2011).
    +
    +
    [2]
    R. Sedgewick, Permutation Generation Methods, ACM Comput. Surv. 9, 137 (1977).
    +
    -
    [2]
    H. Yao, S.-C. Zhang, and S. A. Kivelson, Algebraic Spin Liquid in an Exactly Solvable Spin Model, Phys. Rev. Lett. 102, 217202 (2009).
    +
    [3]
    H. Yao, S.-C. Zhang, and S. A. Kivelson, Algebraic Spin Liquid in an Exactly Solvable Spin Model, Phys. Rev. Lett. 102, 217202 (2009).
    diff --git a/_thesis/toc.html b/_thesis/toc.html index 5f127a6..3c39b94 100644 --- a/_thesis/toc.html +++ b/_thesis/toc.html @@ -13,23 +13,17 @@
  • 3 The Long Range Falicov-Kimball Model
  • 4 The Amorphous Kitaev Model
    • -
  • 5 Conclusion
    • -
  • Appendices
      • diff --git a/assets/thesis/amk_chapter/visual_kitaev_1.svg b/assets/thesis/amk_chapter/visual_kitaev_1.svg index a95c798..dd5fa61 100644 --- a/assets/thesis/amk_chapter/visual_kitaev_1.svg +++ b/assets/thesis/amk_chapter/visual_kitaev_1.svg @@ -6,7 +6,7 @@ version="1.1" id="svg5942" sodipodi:docname="visual_kitaev_1.svg" - inkscape:version="1.2.1 (9c6d41e4, 2022-07-14)" + inkscape:version="1.3-dev (77bc73e, 2022-05-18)" inkscape:export-filename="visual_kitaev_1.pdf" inkscape:export-xdpi="96" inkscape:export-ydpi="96" @@ -30,12 +30,12 @@ inkscape:document-units="pt" showgrid="false" inkscape:zoom="1.2049729" - inkscape:cx="241.49921" + inkscape:cx="230.29564" inkscape:cy="143.15675" inkscape:window-width="1422" inkscape:window-height="819" inkscape:window-x="18" - inkscape:window-y="48" + inkscape:window-y="25" inkscape:window-maximized="0" inkscape:current-layer="layer2" /> The Hamiltonian Loop Operators Plaquette Operators + style="white-space:pre;shape-inside:url(#rect1539);display:inline;fill:#000000;stroke-width:8.91333;-inkscape-font-specification:'STIXGeneral, Normal';font-family:STIXGeneral;font-weight:normal;font-style:normal;font-stretch:normal;font-variant:normal;font-size:12px;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal" /> Bond Operators @@ -4235,26 +4239,28 @@ The Kitavev Model, Visually + y="9.8608837">The Kitaev Model, Visually Bond Types diff --git a/assets/thesis/background_chapter/localisation_radius_vs_length.svg b/assets/thesis/background_chapter/localisation_radius_vs_length.svg index 297c3bf..61f479a 100644 --- a/assets/thesis/background_chapter/localisation_radius_vs_length.svg +++ b/assets/thesis/background_chapter/localisation_radius_vs_length.svg @@ -1,12 +1,12 @@ @@ -1924,18 +1926,18 @@ id="id-8e8d5363-2b8c-4bd6-901c-979a07881cb8" /> @@ -2014,18 +2016,18 @@ id="id-8c34accc-a04a-436e-b07a-12f33964d290" /> @@ -2051,18 +2053,18 @@ id="id-74bb4d22-59a6-477d-9695-5bf37f4e093f" /> @@ -2088,22 +2090,22 @@ id="id-c34cad69-d07b-4f50-be4d-59fa167232f0" /> + style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947" + d="M 78.425001,50.208598 V 100.83214" + id="path15082" + transform="matrix(0.75,0,0,0.75,2.4896826,-4.6681548)" + sodipodi:nodetypes="cc" /> diff --git a/assets/thesis/intro_chapter/venn_diagram.svg b/assets/thesis/intro_chapter/venn_diagram.svg index e59ee6a..35b3052 100644 --- a/assets/thesis/intro_chapter/venn_diagram.svg +++ b/assets/thesis/intro_chapter/venn_diagram.svg @@ -4,10 +4,10 @@ @@ -82,153 +82,182 @@ id="layer2" inkscape:label="Venn" transform="matrix(0.90965985,0,0,0.90965985,-19.730949,-27.726575)"> - - - - - + + + + + + + + transform="translate(-39.713457,-46.524858)" + style="display:inline"> Interacting + id="tspan1">Interacting +Theories + Ideal Gas + id="tspan3">Ideal Gas Band + id="tspan4">Band Insulators + x="142.85497" + y="91.045364" + id="tspan5">Theory + Many Body + id="tspan6">Many Body +Theories Quantum + id="tspan8">Quantum +Theories Hydrogen Atom + style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1837px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;white-space:pre;shape-inside:url(#rect1816);shape-padding:8.33653;display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1" + transform="matrix(1.3114339,0,0,1.3114339,173.84223,125.33066)" + x="22.318253" + y="0">Hydrogen Atom Molecules + style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1837px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, 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