diff --git a/_sass/thesis.scss b/_sass/thesis.scss index a77c544..1148176 100644 --- a/_sass/thesis.scss +++ b/_sass/thesis.scss @@ -35,13 +35,25 @@ main > ul > ul > li { margin-top: 0.5em; } -// Mess with the formatting of the citations +// Pull the citations a little closer in to the previous word +span.citation { + margin-left: -1em; + + a { + text-decoration: none; + color: darkblue; + } +} + +// Mess with the formatting of the bibliography div.csl-entry { margin-bottom: 0.5em; } -// div.csl-entry a { +div.csl-entry a { // text-decoration: none; -// } + text-decoration: none; + color: darkblue; +} div.csl-entry div { display: inline; diff --git a/_thesis/1_Introduction/1_Intro.html b/_thesis/1_Introduction/1_Intro.html index e6fc875..dd83865 100644 --- a/_thesis/1_Introduction/1_Intro.html +++ b/_thesis/1_Introduction/1_Intro.html @@ -204,43 +204,35 @@ image:
-

Interacting Quantum Many -Body Systems

+

Interacting Quantum Many Body Systems

When you take many objects and let them interact together, it is -often simpler to describe the behaviour of the group differently than -one would describe the individual objects. Consider a flock (technically -called a murmuration) of starlings like 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 phenomena is couched in terms of the flock rather than the -individual birds.

-

The behaviours of the flock are an emergent phenomena. The starlings -are only interacting with their immediate six or seven neighbours1. +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 phenomena is couched in terms of the flock +rather than of the individual birds.

+

The behaviours of the flock are an emergent phenomena. The +starlings are only interacting with their immediate six or seven +neighbours  [1,2. This is what a physicist would -call a local interaction. There is much philosophical debate -about how exactly to define emergence2], what a physicist would call a +local interaction. There is much philosophical debate about how +exactly to define emergence  [3,4 but for our purposes it enough -to say that emergence is the fact that the aggregate behaviour of many -interacting objects may be very different from the individual behaviour -of those objects.

+role="doc-biblioref">4] but for our purposes it enough to say +that emergence is the fact that the aggregate behaviour of many +interacting objects may necessitate a description very different from +that of the individual objects.

CC BY-SA 3.0
-

To give another example, our understanding of thermodynamics began -with bulk properties like heat, energy, pressure and temperatureTo 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 particles5]. It was only later that we gained an +understanding of how these properties emerge from microscopic +interactions between very large numbers of particles  [6.

+role="doc-biblioref">6].

Condensed Matter is, at its heart, the study of what behaviours emerge from large numbers of interacting quantum objects at low energy. When these three properties are present together: a large number of @@ -273,60 +265,65 @@ these three ingredients nature builds all manner of weird and wonderful materials.

Historically, we made initial headway in the study of many-body systems, ignoring interactions and quantum properties. The ideal gas law -and the Drude classical electron gas [7 are good examples. Including -interactions into many-body physics leads to the Ising model7] are good examples. Including +interactions into many-body physics leads to the Ising model  [8, Landau theory9 and -the classical theory of phase transitions8], Landau theory  [9] and the classical theory of phase +transitions  [10. In contrast, condensed matter -theory got it state in quantum many-body theory. Bloch’s theorem10]. In contrast, condensed matter +theory got it state in quantum many-body theory. Bloch’s theorem  [11 predicted the properties of +role="doc-biblioref">11] predicted the properties of non-interacting electrons in crystal lattices, leading to band theory. In the same vein, advances were made in understanding the quantum -origins of magnetism, including ferromagnetism and -antiferromagnetism [12.

+role="doc-biblioref">12].

However, at some point we had to start on the interacting quantum -many body systems. Some phenomena cannot be understood without a taking -into account all three effects. The canonical examples are -superconductivity [13, the fractional quantum hall -effect13], the fractional quantum hall effect + [14 and the Mott insulators14] and the Mott insulators  [15,16. We will discuss the latter in -more detail.

-

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. Anderson Insulators have only localised electronic states near -the fermi level and therefore fail the second criteria. We will discuss -Anderson insulators and disorder in a later section.

+role="doc-biblioref">16]. We’ll start by looking at the +latter but shall see that there are many links between three topics.

+

Mott Insulators

+

Mott Insulators are remarkable because their electrical insulator +properties come from electron-electron interactions. 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. Anderson Insulators have only localised electronic +states near the fermi level and therefore fail the second criteria. We +will discuss Anderson insulators and disorder in a later section.

Both band and Anderson insulators occur without electron-electron -interactions. Mott insulators, by contrast, are by these interactions -and hence elude band theory and single-particle methods.

+interactions. Mott insulators, by contrast, require a many body picture +to understand and thus elude band theory and single-particle +methods.

-

Mott Insulators and The -Hubbard Model

The theory of Mott insulators developed out of the observation that many transition metal oxides are erroneously predicted by band theory to -be conductive [17 leading to the suggestion that -electron-electron interactions were the cause of this effect17] leading to the suggestion that +electron-electron interactions were the cause of this effect  [18. Interest grew with the -discovery of high temperature superconductivity in the cuprates in -198618]. Interest grew with the discovery of +high temperature superconductivity in the cuprates in 1986  [19 which is believed to arise as -the result of doping a Mott insulator state19] which is believed to arise as the +result of a doped Mott insulator state  [20.

-

The canonical toy model of the Mott insulator is the Hubbard -model20].

+

The canonical toy model of the Mott insulator is the Hubbard model + [21–23 of 23] of \(1/2\) fermions hopping on the lattice with hopping parameter \(t\) and electron-electron repulsion \(U\)

-

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

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

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 -\(i\). 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. In the interacting limit \(U >> t\) on the other hand, the -system breaks up into a product of local moments, each in one the four -states \(|0\rangle, |\uparrow\rangle, -|\downarrow\rangle, |\uparrow\downarrow\rangle\) depending on the -filing.

+\(i\). 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 Hermition.

+

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. In the interacting limit \(U >> +t\) on the other hand, the system breaks up into a product of +local moments, each in one the four states \(|0\rangle, |\uparrow\rangle, |\downarrow\rangle, +|\uparrow\downarrow\rangle\) depending on the filing.

The Mott insulating phase occurs at half filling \(\mu = \tfrac{U}{2}\) where there is one -electron per lattice site [24. Here the model can be -rewritten in a symmetric form \[ H = -t +role="doc-biblioref">24]. Here the model can be rewritten in +a symmetric form \[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2})\]

The basic reason that the half filled state is insulating seems is 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 Mott insulator. Originally -it was proposed that antiferromagnetic order was a necessary condition -for the Mott insulator transition [25 but later examples were found -without magnetic order cite.

+role="doc-biblioref">25]. However, Mott insulators have been +found  [26,27] without magnetic order. Instead the +local moments may form a highly entangled state known as a quantum spin +liquid, which will be discussed shortly.

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) [26 and dynamical mean-field -theory28] and dynamical mean-field theory + [27. None of these approaches is +role="doc-biblioref">29]. None of these approaches are perfect. Strong correlations are poorly described by the Fermi liquid theory and the 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 [28.

+role="doc-biblioref">30].

From here the discussion will branch two directions. First, we will -discuss a limit of the Hubbard model called the Falikov Kimball Model. -Second, we will go down the rabbit hole of strongly correlated systems -without magnetic order. This will lead us to Quantum spin liquids and -the Kitaev honeycomb model.

-

An exactly solvable model of the Mott Insulator - -demonstrate mott insulator in hubbard model, briefly tease the falikov -kimball model in order to lay the ground work to talk about the falikov -kimball model later

- -

An exactly solvable Quantum Spin Liquid - -relationship between mott insulators and spin liquids: the electrons in -a mott insulator form local moments that normally form an AFM ground -state but if they don’t, due to frustration or other reason, the local -moments may form a QSL at T=0 instead.29, +

The Falikov-Kimball Model

+

Though not the original reason for its introduction, the +Falikov-Kimball (FK) model is the limit of the Hubbard model as the mass +ratio of the spin up and spin down electron is taken to infinity. This +gives a model with two fermion species, one itinerant and one entirely +immobile. The number operators for the immobile fermions are therefore +conserved quantities and can be be treated like classical degrees of +freedom. For our purposes it will be useful to replace the immobile +fermions with a classical Ising background field \(S_i = \pm1\).

+

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

+

Given that the physics of states near the metal-insulator (MI) +transition is still poorly understood  [31,32] the FK model provides a rich test +bed to explore interaction driven MI transition physics. Despite its +simplicity, the model has a rich phase diagram in \(D \geq 2\) dimensions. It shows an Mott +insulator transition even at high temperature, similar to the +corresponding Hubbard Model  [33]. In 1D, the ground state +phenomenology as a function of filling can be rich  [34] but the system is disordered for all +\(T > 0\)  [35]. 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  [36–39].

+

In Chapter 3 I will introduce a generalized FK model in one +dimension. With the addition of long-range interactions in the +background field, the model shows a similarly rich phase diagram. I use +an exact Markov chain Monte Carlo method to map the phase diagram and +compute the energy-resolved localization properties of the fermions. I +then compare the behaviour of this transitionally invariant model to an +Anderson model of uncorrelated binary disorder about a background charge +density wave field which confirms that the fermionic sector only fully +localizes for very large system sizes.

+

An exactly solvable Quantum Spin Liquid

+

To turn to the other key topic of this thesis, we have discussed the +question of the magnetic ordering of local moments in the Mott +insulating state. The local moments may form an AFM ground state. +Alternatively they may fail to order even at zero temperature  [26,30

+role="doc-biblioref">27], giving rise to what is known as a +quantum spin liquid (QSL) state.

+

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

-

kinds of mott insulators: Mott-Heisenberg (AFM order below Néel @@ -507,263 +560,327 @@ designed to fill this gap and present the results.

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

-
+
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diff --git a/_thesis/2_Background/2.2_HKM_Model.html b/_thesis/2_Background/2.2_HKM_Model.html index 3e68040..1e12b28 100644 --- a/_thesis/2_Background/2.2_HKM_Model.html +++ b/_thesis/2_Background/2.2_HKM_Model.html @@ -319,10 +319,10 @@ Majorana \(c_i\) per site.
diff --git a/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html b/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html index 74ac2a0..92ab0d9 100644 --- a/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html +++ b/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html @@ -360,10 +360,10 @@ The fact they’re uncorrelated is key as we’ll see later. Examples of direct sampling methods range from the trivial: take n random bits to generate integers uniformly between 0 and \(2^n\) to more complex methods such as -inverse transform sampling and rejection sampling [1.

+role="doc-biblioref">1].

In physics the distribution we usually want to sample from is the Boltzmann probability over states of the system \(S\): \[ @@ -383,9 +383,9 @@ with system size. Even if we could calculate \(\mathcal{Z}\), sampling from an exponentially large number of options quickly become tricky. This kind of problem happens in many other disciplines too, particularly when -fitting statistical models using Bayesian inference2.

+fitting statistical models using Bayesian inference  [2].

Markov Chains

So what can we do? Well it turns out that if we’re willing to give up in the requirement that the samples be uncorrelated then we can use MCMC @@ -393,11 +393,11 @@ instead.

MCMC defines a weighted random walk over the states \((S_0, S_1, S_2, ...)\), such that in the long time limit, states are visited according to their probability \(p(S)\).\(p(S)\).  [3–5.

+role="doc-biblioref">5].

In a physics context this lets us evaluate any observable with a mean over the states visited by the walk. \[\begin{aligned} @@ -407,9 +407,9 @@ class="math display">\[\begin{aligned}

The choice of the transition function for MCMC is under-determined as one only needs to satisfy a set of balance conditions for which there are many solutions  [6.

+role="doc-biblioref">6].

Application to the FK Model

We will work in the grand canonical ensemble of fixed temperature, chemical potential and volume.

@@ -447,11 +447,11 @@ F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]} expectation values \(\expval{O}\) with respect to some physical system defined by a set of states \(\{x: x \in S\}\) and a free energy \(F(x)\)\(F(x)\)  [7. The thermal expectation value -is defined via a Boltzmann weighted sum over the entire states: 7]. The thermal expectation value is +defined via a Boltzmann weighted sum over the entire states: \[ \begin{aligned} \expval{O} &= \frac{1}{\mathcal{Z}} \sum_{x \in S} O(x) P(x) \\ @@ -526,10 +526,10 @@ P(x) \mathcal{T}(x \rightarrow x') = P(x') \mathcal{T}(x' \rightarrow x) \] % In practice most algorithms are constructed to satisfy detailed balance though there are arguments that relaxing the condition -can lead to faster algorithms [8.

+role="doc-biblioref">8].

The goal of MCMC is then to choose \(\mathcal{T}\) so that it has the desired thermal distribution \(P(x)\) as its @@ -558,10 +558,10 @@ x_{i}\). Now \(\mathcal{T}(x\to x')

The Metropolis-Hasting algorithm is a slight extension of the original Metropolis algorithm that allows for non-symmetric proposal distributions $q(xx’) q(x’x) $. It can be derived starting from detailed -balance [7: 7]: \[\begin{aligned} P(x)\mathcal{T}(x \to x') &= P(x')\mathcal{T}(x' \to x) \\ @@ -671,11 +671,11 @@ problematic because it means very few new samples will be generated. If it is too high it implies the steps are too small, a problem because then the walk will take longer to explore the state space and the samples will be highly correlated. Ideal values for the acceptance rate -can be calculated under certain assumptions [9. Here we monitor the acceptance -rate and if it is too high we re-run the MCMC with a modified proposal +role="doc-biblioref">9]. Here we monitor the acceptance rate +and if it is too high we re-run the MCMC with a modified proposal distribution that has a chance to propose moves that flip multiple sites at a time.

In addition we exploit the particle-hole symmetry of the problem by @@ -686,10 +686,10 @@ produce a state at or near the energy of the current one.

The matrix diagonalisation is the most computationally expensive step of the process, a speed up can be obtained by modifying the proposal distribution to depend on the classical part of the energy, a trick -gleaned from Ref. [7: \[ +role="doc-biblioref">7]: \[ \begin{aligned} q(k \to k') &= \min\left(1, e^{\beta (H^{k'} - H^k)}\right) \\ @@ -700,12 +700,11 @@ without performing the diagonalisation at no cost to the accuracy of the MCMC method.

An extension of this idea is to try to define a classical model with a similar free energy dependence on the classical state as the full -quantum, Ref. [10 does this with restricted -Boltzmann machines whose form is very similar to a classical spin -model.

+role="doc-biblioref">10] does this with restricted Boltzmann +machines whose form is very similar to a classical spin model.

Scaling

In order to reduce the effects of the boundary conditions and the finite size of the system we redefine and normalise the coupling matrix @@ -726,12 +725,12 @@ central moments of the order parameter m: \[m = \sum_i (-1)^i (2n_i - 1) / N\] % The Binder cumulant evaluated against temperature can be used as a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different -system sizes, a crossing indicates the location of a critical point [11,musialMonteCarloSimulations2002?.

+role="doc-biblioref">musialMonteCarloSimulations2002?].

Markov Chain Monte-Carlo in Practice

Quick Intro to MCMC

@@ -758,13 +757,13 @@ very expensive operation!~\footnote{The effort involved in exact diagonalisation scales like \(N^2\) for systems with a tri-diagonal matrix representation (open boundary conditions and nearest neighbour hopping) and like \(N^3\) for a generic matrix\(N^3\) for a generic matrix  [12,13.

+role="doc-biblioref">13].

c

MCMC sidesteps these issues by defining a random walk that focuses on the states with the greatest Boltzmann weight. At low temperatures this @@ -878,10 +877,10 @@ auto-correlation time \(\tau(O)\) informally as the number of MCMC samples of some observable O that are statistically equal to one independent sample or equivalently as the number of MCMC steps after which the samples are correlated below some -cutoff, see [14 for a more rigorous definition +role="doc-biblioref">14] for a more rigorous definition involving a sum over the auto-correlation function. The auto-correlation time is generally shorter than the convergence time so it therefore makes sense from an efficiency standpoint to run a single walker for @@ -960,28 +959,28 @@ than the current state.

Two Step Trick

Here, we incorporate a modification to the standard Metropolis-Hastings algorithm  [15 gleaned from Krauth 15] gleaned from Krauth  [7.

+role="doc-biblioref">7].

In our computations 16 we employ a modification of the -algorithm which is based on the observation that the free energy of the -FK system is composed of a classical part which is much quicker to -compute than the quantum part. Hence, we can obtain a computational -speedup by first considering the value of the classical energy -difference \(\Delta H_s\) and rejecting -the transition if the former is too high. We only compute the quantum -energy difference \(\Delta F_c\) if the -transition is accepted. We then perform a second rejection sampling step -based upon it. This corresponds to two nested comparisons with the -majority of the work only occurring if the first test passes and has the -acceptance function \[\mathcal{A}(a \to b) = -\min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta +data-cites="hodsonMCMCFKModel2021"> [16] we +employ a modification of the algorithm which is based on the observation +that the free energy of the FK system is composed of a classical part +which is much quicker to compute than the quantum part. Hence, we can +obtain a computational speedup by first considering the value of the +classical energy difference \(\Delta +H_s\) and rejecting the transition if the former is too high. We +only compute the quantum energy difference \(\Delta F_c\) if the transition is accepted. +We then perform a second rejection sampling step based upon it. This +corresponds to two nested comparisons with the majority of the work only +occurring if the first test passes and has the acceptance function \[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta +\Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]

For the model parameters used in Fig. 2, we find @@ -1008,9 +1007,9 @@ distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model  [10. The RBMs are chosen as a +role="doc-biblioref">10]. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is @@ -1021,11 +1020,11 @@ the two step method

Given a MCMC algorithm with target distribution \(\pi(a)\) and transition function \(\mathcal{T}\) the detailed balance -condition is sufficient (along with some technical constraints [5) to guarantee that in the long -time limit the algorithm produces samples from 5]) to guarantee that in the long time +limit the algorithm produces samples from \(\pi\). \[\pi(a)\mathcal{T}(a \to b) = \pi(b)\mathcal{T}(b \to a)\]

@@ -1141,10 +1140,10 @@ for the additional complexity it would require.

Inverse Participation Ratio

The inverse participation ratio is defined for a normalised wave function \(\psi_i = \psi(x_i), \sum_i -\abs{\psi_i}^2 = 1\) as its fourth moment as its fourth moment  [17: \[ +role="doc-biblioref">17]: \[ P^{-1} = \sum_i \abs{\psi_i}^4 \] % It acts as a measure of the portion of space occupied by the wave function. For localised states it will be independent of system @@ -1155,11 +1154,10 @@ fractal dimensionality \(d > d* > P(L) \goeslike L^{d*} \] % For extended states \(d* = 0\) while for localised ones \(d* = -0\). In this work we take use an energy resolved IPR. In this work we take use an energy resolved IPR  [18: \[ +role="doc-biblioref">18]: \[ DOS(\omega) = \sum_n \delta(\omega - \epsilon_n) IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_{n,i}}^4 @@ -1518,148 +1516,141 @@ class="sourceCode python">

-
+
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\(N^3\) for a generic matrix  [12,13.13].↩︎

  • or, in the general case, any desired distribution. MCMC has found a lot of use in sampling from the @@ -1689,9 +1680,9 @@ role="doc-backlink">↩︎

  • or equivalently as the number of MCMC steps after which the samples are correlated below some cutoff, see  [14 for a more rigorous definition +role="doc-biblioref">14] for a more rigorous definition involving a sum over the auto-correlation function.↩︎

    • diff --git a/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html b/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html index 1e7f656..fc95765 100644 --- a/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html +++ b/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html @@ -327,9 +327,9 @@ constant \(U=5\) and constant \(J=5\), respectively. We determined the transition temperatures from the crossings of the Binder cumulants \(B_4 = \tex{m^4}/\tex{m^2}^2\)  [1. For a representative set of +role="doc-biblioref">1]. For a representative set of parameters, Fig. [1c] shows the order parameter \(\tex{m}^2\). Fig. [

    Our main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref.  [2, we can distinguish between the -Mott and Anderson insulating phases. The former is characterised by a -gapped DOS in the absence of a CDW. Thus, the opening of a gap for large -\(U\) is distinct from the gap-opening +role="doc-biblioref">2], we can distinguish between the Mott +and Anderson insulating phases. The former is characterised by a gapped +DOS in the absence of a CDW. Thus, the opening of a gap for large \(U\) is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below \(T_c\), see also Fig. [

    At infinite temperature, all the spin configurations become equally likely and the fermionic model reduces to one of binary uncorrelated disorder in which all eigenstates are Anderson localised  [3. An Anderson localised state -centered around \(r_0\) has magnitude -that drops exponentially over some localisation length 3]. An Anderson localised state centered +around \(r_0\) has magnitude that drops +exponentially over some localisation length \(\xi\) i.e \(|\psi(r)|^2 \sim \exp{-\abs{r - r_0}/\xi}\). Calculating \(\xi\) @@ -417,12 +416,12 @@ additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition  [4,5. The thermal defects of the CDW -phase lead to a binary disorder potential with a finite correlation -length, which in principle could result in delocalized eigenstates.

    +role="doc-biblioref">5]. The thermal defects of the CDW phase +lead to a binary disorder potential with a finite correlation length, +which in principle could result in delocalized eigenstates.

    The key question for our system is then: How is the \(T=0\) CDW phase with fully delocalized fermionic states connected to the fully localized phase at high @@ -488,7 +487,7 @@ alt="The DOS (a) and scaling exponent \tau (b) as a function of energy for the C

    +alt="Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation  [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" />
    -

    More general argumentsMore general arguments  [4,5 imply that 5] imply that \(det(Q^u) \prod -i u_{ij}\) has an interesting relationship to the topological fluxes. In the non-Abelian phase, we expect that it will change sign in exactly one of the four @@ -838,8 +837,8 @@ definition, the vortex free sector.

    On the Honeycomb, Lieb’s theorem implies that the ground state corresponds to the state where all \(u_{jk} = 1\). This implies that the flux free sector is the ground state -sector6.

    +sector  [6].

    Lieb’s theorem does not generalise easily to the amorphous case. However, we can get some intuition by examining the problem that will lead to a guess for the ground state. We will then provide numerical @@ -919,12 +918,11 @@ i)^{n_{\mathrm{sides}}}, class="math inline">\(n_{\mathrm{sides}}\) is the number of edges that form each plaquette and the choice of sign gives a twofold chiral ground state degeneracy.

    -

    This conjecture is consistent with Lieb’s theorem on regular -lattices6 and -is supported by numerical evidence. As noted before, any flux that -differs from the ground state is an excitation which we call a -vortex.

    +

    This conjecture is consistent with Lieb’s theorem on regular lattices + [6] and is +supported by numerical evidence. As noted before, any flux that differs +from the ground state is an excitation which we call a vortex.

    Finite size effects

    This guess only works for larger lattices. To rigorously test it, we would want to directly enumerate the chiral degeneracy which arises because the global sign of the odd plaquettes does not matter.

    This happens because we have broken the time reversal symmetry of the -original model by adding odd plaquettes [7–14.

    +href="#ref-WangHaoranPRB2021" role="doc-biblioref">14].

    Similarly to the behaviour of the original Kitaev model in response to a magnetic field, we get two degenerate ground states of different handedness. Practically speaking, one ground state is related to the other by inverting the imaginary \(\phi\) fluxes\(\phi\) fluxes  [8.

    +role="doc-biblioref">8].

    Phases of the Kitaev Model

    discuss the Abelian A phase / toric code phase / anisotropic phase

    @@ -1114,190 +1111,185 @@ and construct the set \((+1, +1), (+1, -1),
    +alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make  [15]." /> +would be a lot easier to make  [15].
    -

    However, in the non-Abelian phase we have to wrangle with -monodromyHowever, in the non-Abelian phase we have to wrangle with monodromy + [4,5. Monodromy is the behaviour of +role="doc-biblioref">5]. Monodromy is the behaviour of objects as they move around a singularity. This manifests here in that the identity of a vortex and cloud of Majoranas can change as we wind them around the torus in such a way that, rather than annihilating to the vacuum, we annihilate them to create an excited state instead of a ground state. This means that we end up with only three degenerate ground states in the non-Abelian phase \((+1, -+1), (+1, -1), (-1, +1)\)  [3,16. Concretely, this is because -the projector enforces both flux and fermion parity. When we wind a -vortex around both non-contractible loops of the torus, it flips the -flux parity. Therefore, we have to introduce a fermionic excitation to -make the state physical. Hence, the process does not give a fourth -ground state.

    +role="doc-biblioref">16]    . Concretely, this is because the +projector enforces both flux and fermion parity. When we wind a vortex +around both non-contractible loops of the torus, it flips the flux +parity. Therefore, we have to introduce a fermionic excitation to make +the state physical. Hence, the process does not give a fourth ground +state.

    Recently, the topology has notably gained interest because of proposals to use this ground state degeneracy to implement both -passively fault tolerant and actively stabilised quantum -computations [1,17,18.

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    diff --git a/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html b/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html index a6cec0c..1c3a2c2 100644 --- a/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html +++ b/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html @@ -245,11 +245,11 @@ id="toc-open-boundary-conditions">Open boundary conditions guidance from Willian and Johannes. The project grew out of an interest Gino, Peru and I had in studying amorphous systems, coupled with Johannes’ expertise on the Kitaev model. The idea to use voronoi -partitions came from [1 and Gino did the implementation -of this. The idea and implementation of the edge colouring using SAT +role="doc-biblioref">1] and Gino did the implementation of +this. The idea and implementation of the edge colouring using SAT solvers, the mapping from flux sector to bond sector using A* search were both entirely my work. Peru came up with the ground state conjecture and implemented the local markers. Gino and I did much of the @@ -289,11 +289,11 @@ material. Candidate materials, such as \(\alpha\mathrm{-RuCl}_3\), are known to have sufficiently strong spin-orbit coupling and the correct lattice structure to behave according to the Kitaev Honeycomb model with small -corrections [2,trebstKitaevMaterials2022?.

    +role="doc-biblioref">trebstKitaevMaterials2022?].

    expand later: Why do we need spin orbit coupling and what will the corrections be?

    Second, its ground state is the canonical example of the long sought @@ -301,17 +301,17 @@ after quantum spin liquid state. Its excitations are anyons, particles that can only exist in two dimensions that break the normal fermion/boson dichotomy. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to -achieve noise tolerant quantum computations [3.

    +role="doc-biblioref">3].

    Third, and perhaps most importantly, this model is a rare many body interacting quantum system that can be treated analytically. It is exactly solvable. We can explicitly write down its many body ground -states in terms of single particle states [4. The solubility of the Kitaev +role="doc-biblioref">4]. The solubility of the Kitaev Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical @@ -326,9 +326,9 @@ lattices.

    look at the gauge symmetries of the model as well as its solution via a transformation to a Majorana hamiltonian. This discussion shows that, for the the model to be solvable, it needs only be defined on a -trivalent, tri-edge-colourable lattice5.

    +trivalent, tri-edge-colourable lattice  [5].

    The methods section discusses how to generate such lattices and colour them. It also explain how to map back and forth between configurations of the gauge field and configurations of the gauge @@ -512,12 +512,11 @@ on site \(j\) and \(\langle j,k\rangle_\alpha\) is a pair of nearest-neighbour indices connected by an \(\alpha\)-bond with exchange coupling \(J^\alpha\)\(J^\alpha\)  [4. For notational brevity, it is -useful to introduce the bond operators \(K_{ij} = +role="doc-biblioref">4]. For notational brevity, it is useful +to introduce the bond operators \(K_{ij} = \sigma_j^{\alpha}\sigma_k^{\alpha}\) where \(\alpha\) is a function of \(i,j\) that picks the correct bond type.

    @@ -744,10 +743,9 @@ theory of the Majorana Hamiltonian further.

    u_{ij} c_i c_j\] in which most of the Majorana degrees of freedom have paired along bonds to become a classical gauge field \(u_{ij}\). What follows is relatively -standard theory for quadratic Majorana Hamiltonians6.

    +standard theory for quadratic Majorana Hamiltonians  [6].

    Because of the antisymmetry of the matrix with entries \(J^{\alpha} u_{ij}\), the eigenvalues of the Hamiltonian \(\tilde{H}_u\) come in @@ -865,52 +863,49 @@ which we set to 1 when calculating the projector.

    anyway, an arbitrary pairing of the unpaired \(b^\alpha\) operators could be performed. </i,j></i,j>

    -
    +
    -
    1.
    Marsal, Q., Varjas, D. & Grushin, A. G. [1]
    Q. +Marsal, D. Varjas, and A. G. Grushin, Topological Weaire–Thorpe -models of amorphous matter. Proceedings of the National Academy -of Sciences 117, 30260–30265 (2020).
    +Models of Amorphous Matter, Proceedings of the National Academy +of Sciences 117, 30260 (2020).
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    Banerjee, A. et al. [2]
    A. +Banerjee et al., Proximate Kitaev Quantum Spin -Liquid Behaviour in {\alpha}-RuCl$_3$. Nature Mater -15, 733–740 (2016).
    +Liquid Behaviour in {\Alpha}-RuCl$_3$, Nature Mater +15, 733 (2016).
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    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 ed3adba..0265f41 100644 --- a/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html +++ b/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html @@ -234,20 +234,20 @@ Markers

    Methods

    The practical implementation of what is described in this section is available as a Python package called Koala (Kitaev On Amorphous -LAttices) [tomImperialCMTHKoalaFirst2022?. +role="doc-biblioref">tomImperialCMTHKoalaFirst2022?]. All results and figures were generated with Koala.

    Voronisation

    To study the properties of the amorphous Kitaev model, we need to sample from the space of possible trivalent graphs.

    -

    A simple method is to use a Voronoi partition of the torusA simple method is to use a Voronoi partition of the torus  [1–3. We start by sampling seed +role="doc-biblioref">3]. We start by sampling seed points uniformly (or otherwise) on the torus. Then, we compute the partition of the torus into regions closest (with a Euclidean metric) to each seed point. The straight lines (if the torus is flattened out) at @@ -259,23 +259,23 @@ the graph is embedded into the plane. It is also trivalent in that every vertex is connected to exactly three edges cite.

    Ideally, we would sample uniformly from the space of possible trivalent graphs. Indeed, there has been some work on how to do this -using a Markov Chain Monte Carlo approach [4. However, it does not guarantee -that the resulting graph is planar, which we must ensure so that the -edges can be 3-coloured.

    -

    In practice, we use a standard algorithm4]. However, it does not guarantee that +the resulting graph is planar, which we must ensure so that the edges +can be 3-coloured.

    +

    In practice, we use a standard algorithm  [5 from Scipy5] from Scipy  [6 which computes the Voronoi -partition of the plane. To compute the Voronoi partition of the torus, -we take the seed points and replicate them into a repeating grid. This -will be either 3x3 or, for very small numbers of seed points, 5x5. Then, -we identify edges in the output to construct a lattice on the torus.

    +role="doc-biblioref">6] which computes the Voronoi partition +of the plane. To compute the Voronoi partition of the torus, we take the +seed points and replicate them into a repeating grid. This will be +either 3x3 or, for very small numbers of seed points, 5x5. Then, we +identify edges in the output to construct a lattice on the torus.

    This problem must be distinguished from that considered by the famous -four-colour theorem7. The 4-colour theorem is -concerned with assigning colours to the vertices of a -graph, such that no vertices that share an edge have the same colour. -Here we are concerned with an edge colouring.

    +four-colour theorem  [7]. +The 4-colour theorem is concerned with assigning colours to the +vertices of a graph, such that no vertices that share +an edge have the same colour. Here we are concerned with an edge +colouring.

    The four-colour theorem applies to planar graphs, those that can be embedded onto the plane without any edges crossing. Here we are concerned with Toroidal graphs, which can be embedded onto the torus without any edges crossing. In fact, toroidal graphs require up to seven -colours [8. The complete graph 8]. The complete graph \(K_7\) is a good example of a toroidal graph that requires seven colours.

    \(\Delta + 1\) colours are enough to edge-colour any graph. An \(\mathcal{O}(mn)\) algorithm exists to do it for a graph with \(m\) edges and \(n\) vertices\(n\) vertices  [9. Restricting ourselves to graphs -with \(\Delta = 3\) like ours, those -can be four-edge-coloured in linear time9]. Restricting ourselves to graphs with +\(\Delta = 3\) like ours, those can be +four-edge-coloured in linear time  [10.

    +role="doc-biblioref">10].

    However, three-edge-colouring them is more difficult. Cubic, planar, bridgeless graphs can be three-edge-coloured if and only if they can be -four-face-coloured11. An \(\mathcal{O}(n^2)\) algorithm exists -here [11]. An \(\mathcal{O}(n^2)\) algorithm exists here + [12. However, it is not clear -whether this extends to cubic, toroidal bridgeless -graphs.

    +role="doc-biblioref">12]. However, it is not clear whether +this extends to cubic, toroidal bridgeless graphs.

    \(x_1\) or not \(x_3\) is true”, and looks for an assignment \(x_i \in {0,1}\) that -satisfies all the statements13.

    +satisfies all the statements  [13].

    General purpose, high performance programs for solving SAT problems -have been an area of active research for decades [14. 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-Complete14]. 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  [15,16.

    +role="doc-biblioref">16].

    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 @@ -495,9 +494,9 @@ could be used to speed up its solution, using a SAT solver appears to be a reasonable first method to try. As will be discussed later, this turned out to work well enough and looking for a better solution was not necessary.

    -

    We use a solver called MiniSAT17. Like most modern SAT solvers, +

    We use a solver called MiniSAT  [17]. Like most modern SAT solvers, MiniSAT requires the input problem to be specified in Conjunctive Normal Form (CNF). CNF requires that the constraints be encoded as a set of clauses of the form  [18,19. Perhaps

    +role="doc-biblioref">19]. Perhaps

    Expand on definition here

    Discuss link between Chern number and Anyonic Statistics

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    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 48a123d..ede2dfd 100644 --- a/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html +++ b/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html @@ -249,10 +249,10 @@ ground state flux sector is correct. We will do this by enumerating all the flux sectors of many separate system realisations. However there are some issues we will need to address to make this argument work.

    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 +role="doc-biblioref">1] so we would like to perform our analysis for very large lattices. However for an amorphous system with \(N\) plaquettes, \(2N\) edges and

    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 plaquettes2.

    +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 +href="#ref-Peri2020" role="doc-biblioref">4]. This spontaneously broken symmetry avoids the need to explicitly break TRS with a magnetic field term as is done in the original honeycomb model.

    @@ -348,12 +348,11 @@ straight lines \(|J^x| = |J^y| + class="math inline">\(x,y,z\), shown as dotted line on ~1 (Right). We find that on the amorphous lattice these boundaries exhibit an inward curvature, similar to -honeycomb Kitaev models with flux5 or bond6 -disorder.

    +honeycomb Kitaev models with flux  [5] or bond  [6] disorder.

    \(0\) to \(\pm later I’ll double check this with finite size scaling.

    The next question is: do these phases support excitations with Abelian or non-Abelian statistics? To answer that we turn to Chern -numbers [7–9. As discussed earlier the Chern +role="doc-biblioref">9]. 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 @@ -400,28 +399,27 @@ to its Chern number [citation]. However the Chern number is only defined for the translation invariant case because it relies on integrals defined in k-space.

    A family of real space generalisations of the Chern number that work -for amorphous systems exist called local topological markers [10–12 and indeed Kitaev defines one -in his original paper on the model12] and indeed Kitaev defines one in his +original paper on the model  [1.

    -

    Here we use the crosshair marker of13 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 is typically exponentially localised -[cite]. 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

    +role="doc-biblioref">1]    .

    +

    Here we use the crosshair marker of  [13] 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 is typically exponentially localised [cite]. 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 ( @@ -440,54 +438,51 @@ 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 [14.

    +role="doc-biblioref">14].

    The B phase has \(\nu=\pm1\) so is a non-Abelian chiral spin liquid (CSL) similar to that of the -Yao-Kivelson model3. The CSL state is the the -magnetic analogue of the fractional quantum Hall state -[cite]. Hereafter we focus our attention on this -phase.

    +Yao-Kivelson model  [3]. +The CSL state is the the magnetic analogue of the fractional quantum +Hall state [cite]. Hereafter we focus our attention on +this phase.

    +alt="Figure 2: (Center) The crosshair marker  [13], 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 = \pm 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." /> +crosshair marker  [13], 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 \(\nu = \pm 0\) +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 conditions15. fig.  [15]. 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 @@ -522,35 +517,34 @@ states. 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 phasethermal metal phase  [16.

    +role="doc-biblioref">16].

    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 anyon17. The interactions between these +an Ising-type non-Abelian anyon  [17]. The interactions between these anyons are oscillatory similar to the RKKY exchange and decay -exponentially with separation [18–20. At sufficient density, the -anyons hybridise to a macroscopically degenerate state known as -thermal metal18. 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.

    +role="doc-biblioref">20]. At sufficient density, the anyons +hybridise to a macroscopically degenerate state known as thermal +metal  [18]. 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 [16 but in order to avoid that +role="doc-biblioref">16] but in order to avoid that complexity in the current work we instead opted to use vortex density \(\rho\) as a proxy for temperature.

    @@ -641,11 +635,11 @@ 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 [16,21.

    +role="doc-biblioref">21].

    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 @@ -656,21 +650,21 @@ field.

    src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg" data-short-caption="Distinctive Oscillations in the Density of States" style="width:100.0%" -alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state16,21. (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." /> +alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [16,21]. (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." /> +role="doc-biblioref">21]. (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.

    Conclusion

    @@ -719,46 +713,45 @@ Realisations and Signatures

    The obvious question is whether amorphous Kitaev materials could be physically realised.

    Most crystals can as exists in a metastable amorphous state if they -are cooled rapidly, freezing them into a disordered configuration [22–24. -Indeed quenching has been used by humans to control the hardness of -steel or iron for thousands of years. It would therefore be interesting -to study amorphous version of candidate Kitaev materials24]. Indeed +quenching has been used by humans to control the hardness of steel or +iron for thousands of years. It would therefore be interesting to study +amorphous version of candidate Kitaev materials  [trebstKitaevMaterials2022? +role="doc-biblioref">trebstKitaevMaterials2022?] such as \(\alpha-\textrm{RuCl}_3\) to see whether they maintain even approximate fixed coordination number -locally as is the case with amorphous Silicon and Germanium [25,26.

    +role="doc-biblioref">26].

    Looking instead at more engineered realisation, metal organic frameworks have been shown to be capable of forming amorphous -lattices 27 and there are recent proposals -for realizing strong Kitaev interactions  [27] and +there are recent proposals for realizing strong Kitaev +interactions  [28 as well as reports of QSL +role="doc-biblioref">28] as well as reports of QSL behavior  [29.

    +role="doc-biblioref">29].

    Generalisations

    The model presented here could be generalized in several ways.

    First, it would be interesting to study the stability of the chiral amorphous Kitaev QSL with respect to perturbations  [30–34.

    +href="#ref-Winter2016" role="doc-biblioref">34].

    Second, one could investigate whether a QSL phase may exist for for other models defined on amorphous lattices. For example, in real materials, there will generally be an additional small Heisenberg term @@ -767,398 +760,382 @@ j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} + \sigma_j\sigma_k\] With a view to more realistic prospects of observation, it would be interesting to see if the properties of the Kitaev-Heisenberg model generalise from the honeycomb to the amorphous -case[31;33;35;36; [31];  [33];  [35];  [36];  [37;].

    +role="doc-biblioref">37];].

    Finally it might be possible to look at generalizations to higher-spin models or those on random networks with different -coordination numbers [2,38–47

    +href="#ref-Wu2009" role="doc-biblioref">47]

    Overall, there has been surprisingly little research on amorphous quantum many body phases albeit material candidates aplenty. We expect our exact chiral amorphous spin liquid to find many generalisation to realistic amorphous quantum magnets and beyond.

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