I would like to thank my supervisor, Professor Johannes Knolle and co-supervisor Professor Derek Lee for guidance and support during this long process.
diff --git a/_thesis/1_Introduction/1_Intro.html b/_thesis/1_Introduction/1_Intro.html index dd83865..1b3987e 100644 --- a/_thesis/1_Introduction/1_Intro.html +++ b/_thesis/1_Introduction/1_Intro.html @@ -199,15 +199,42 @@ image: -{% include header.html %} + + +{% capture tableOfContents %} ++Contents: + +{% endcapture %} + + +{% include header.html extra=tableOfContents %}
+Contents: +
-
+
- Markov Chain Monte Carlo
+
-
+
- Sampling +
- Markov +Chains +
- Application to the FK Model + +
- The Metropolis-Hasting +Algorithm +
- Metropolis-Hastings +
- Convergence, +Auto-correlation and Binning +
- Applying MCMC to the FK +model +
- Proposal Distributions +
- Perturbation +MCMC +
- Scaling +
- Binder +Cumulants +
- Markov Chain Monte-Carlo +in Practice + +
- Two Step +Trick +
- Detailed Balance for +the two step method + +
- Diagnostics of Localisation + +
- Markov Chain Monte-Carlo +
- Convergence Time +
- Auto-correlation Time +
- The Metropolis-Hastings +Algorithm +
- Choosing the proposal +distribution +
- Two Step +Trick +
+
Markov Chain Monte Carlo
Sampling
Markov Chain Monte Carlo (MCMC) is a useful method whenever we have a @@ -360,7 +449,7 @@ 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].
@@ -383,7 +472,7 @@ 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 inference [2].Markov Chains
@@ -393,7 +482,7 @@ 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–\(\expval{O}\) 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 @@ -526,7 +615,7 @@ 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].
@@ -558,7 +647,7 @@ 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]: [9]. Here we monitor the acceptance rate @@ -686,7 +775,7 @@ 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]: \[ @@ -700,7 +789,7 @@ 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 @@ -725,8 +814,8 @@ 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,\(N^2\) 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 [\(\tau(O)\) 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 @@ -1020,7 +1109,7 @@ 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 @@ -1140,7 +1229,7 @@ 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]: \[
@@ -1154,7 +1243,7 @@ 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]: \[
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 fc95765..3fddc7d 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
@@ -262,10 +262,31 @@ image:
-{% include header.html %}
+
+
+{% capture tableOfContents %}
+ The bond operators \(u_{ij}\) are
useful because they label a bond sector and \(2^2 =
4\) topological sectors. The topological sector forms the basis of proposals to construct
topologically protected qubits since the four sectors can only be mixed
-by a highly non-local perturbations [1]. \[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i
\prod_i^{2N} b^y_i \prod_i^{2N} b^z_i \prod_i^{2N} c_i\] The product over \(c_i\) operators
-reduces to a determinant of the Q matrix and the fermion parity, see
- [2]. The only difference from the
@@ -702,7 +758,7 @@ 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 \(n_i\). As discussed in [2], \([\hat{\pi}, D_i] = 0\). 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]. More general arguments More general arguments [4,
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
-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
@@ -918,8 +974,8 @@ i)^{n_{\mathrm{sides}}},
class="math inline">\(n_{\mathrm{sides}}\) This conjecture is consistent with Lieb’s theorem on regular lattices
- [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.
+Contents:
+
+
+{% endcapture %}
+
+
+{% include header.html extra=tableOfContents %}
-{% include header.html %}
+
+
+{% capture tableOfContents %}
+
+Contents:
+
+
+{% endcapture %}
+
+
+{% include header.html extra=tableOfContents %}
Gauge Fields
This happens because we have broken the time reversal symmetry of the -original model by adding odd plaquettes [7–14].
@@ -981,7 +1037,7 @@ href="#ref-WangHaoranPRB2021" role="doc-biblioref">14]. 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]. @@ -1111,18 +1167,18 @@ and construct the set \((+1, +1), (+1, -1),
However, in the non-Abelian phase we have to wrangle with monodromy
-However, in the non-Abelian phase we have to wrangle with
+monodromy [4,\((+1,
-+1), (+1, -1), (-1, +1)\) [3,
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,
-{% include header.html %}
+
+
+{% capture tableOfContents %}
+ The material in this chapter expands on work presented in Insert citation of amorphous Kitaev paper here 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
@@ -326,7 +368,7 @@ lattices.
+Contents:
+
+
+{% endcapture %}
+
+
+{% include header.html extra=tableOfContents %}
Contributions
The methods section discusses how to generate such lattices and @@ -512,7 +554,7 @@ 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 @@ -743,7 +785,7 @@ 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 Hamiltonians [6].Because of the antisymmetry of the matrix with entries
-{% include header.html %}
+
+
+{% capture tableOfContents %}
+ The practical implementation of what is described in this section is
available as a Python package called Koala (Kitaev On Amorphous
-LAttices) [tomImperialCMTHKoalaFirst2022?].
@@ -242,7 +277,7 @@ All results and figures were generated with Koala. 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 torus A simple method is to use a Voronoi partition of the torus [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 algorithm In practice, we use a standard algorithm [5] from Scipy 5] from Scipy [6] which computes the Voronoi partition
@@ -368,7 +403,7 @@ onto the plane without any edges crossing. Bridgeless graphs do not
contain any edges that, when removed, would partition the graph into
disconnected components. This problem must be distinguished from that considered by the famous
-four-colour theorem [7].
The 4-colour theorem is concerned with assigning colours to the
@@ -379,7 +414,7 @@ colouring.
+Contents:
+
+
+{% endcapture %}
+
+
+{% include header.html extra=tableOfContents %}
Methods
Voronisation
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-coloured [11]. An \(\mathcal{O}(n^2)\) algorithm exists here - [\(\mathcal{O}(n^2)\) algorithm exists +here [12]. However, it is not clear whether this extends to cubic, toroidal bridgeless graphs.
@@ -466,17 +501,17 @@ solver. A SAT problem is a set of statements about some number of boolean variables , such as “\(x_1\) or not \(x_3\) is true”, and looks for an assignment \(x_i \in {0,1}\) that -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-Complete [
- We use a solver called Here I will discuss the numerical evidence that our guess for the
@@ -249,7 +295,7 @@ 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
@@ -308,7 +354,7 @@ relatively regular pattern for the imaginary fluxes with only a global
two-fold chiral degeneracy. 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]. So we have flux sectors that come in degenerate pairs, where time
@@ -348,9 +394,9 @@ 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 flux [5] or bond 5 The next question is: do these phases support excitations with
Abelian or non-Abelian statistics? To answer that we turn to Chern
-numbers [7–[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 model [1]. Here we use the crosshair marker of Here we use the crosshair marker of [13] because it works well on smaller
systems. We calculate the projector \(P =
@@ -438,13 +484,14 @@ 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]. 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
@@ -455,9 +502,9 @@ this phase. Chiral Spin Liquids support topological protected edge modes on open
-boundary conditions [15]. fig. 3 shows the probability density of one such
@@ -517,31 +564,31 @@ 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 phase thermal metal phase [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 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
-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
@@ -635,7 +682,7 @@ 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,
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 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." />
+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." />
MiniSAT
We use a solver called MiniSAT
[17]. Like most modern SAT solvers,
MiniSAT
requires the input problem to be specified in
@@ -554,7 +589,7 @@ 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 [18,
-{% include header.html %}
+
+
+{% capture tableOfContents %}
+
+Contents:
+
+
+{% endcapture %}
+
+
+{% include header.html extra=tableOfContents %}
Results
The Ground State Flux Sector
+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." />
Edge Modes