Landau theory characterises phases of matter as inextricably linked -to the emergence of long range order via a spontaneously broken -symmetry. The fractional quantum Hall (FQH) state, discovered in the -1980s is an explicit example of an electronic system that falls outside -of the Landau paradigm. FQH systems exhibit fractionalised excitations -linked to their ground state having long range entanglement and -non-trivial topological properties [Landau-Ginzburg-Wilson theory characterises phases of matter as +inextricably linked to the emergence of long range order via a +spontaneously broken symmetry. The fractional quantum Hall (FQH) state, +discovered in the 1980s is an explicit example of an electronic system +that falls outside of the Landau-Ginzburg-Wilson paradigm. FQH systems +exhibit fractionalised excitations linked to their ground state having +long range entanglement and non-trivial topological properties [40]. Quantum spin liquids are the analogous phase of matter for spin systems. Remarkably the existence of @@ -595,33 +598,40 @@ problem via a mapping to Majorana fermions which yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state, it supports a rich phase diagram hosting gapless, -Abelian and non-Abelian phases and a finite temperature phase transition -to a thermal metal state [51] and a finite temperature phase +transition to a thermal metal state [51]. It has also been proposed that it -could be used to support topological quantum computing 52]. It been proposed that its +non-Abelian excitations could be used to support robust topological +quantum computing [ [53]; [52].
+role="doc-biblioref">54]; +nayakNonAbelianAnyonsTopological2008].It is by now understood that the Kitaev model on any tri-coordinated \(z=3\) graph has conserved plaquette operators and local symmetries [53,54] which allow a mapping onto effective +role="doc-biblioref">55,56] which allow a mapping onto effective free Majorana fermion problems in a background of static \(\mathbb Z_2\) fluxes [55–58]. +href="#ref-Nussinov2009" role="doc-biblioref">57–60]. However, depending on lattice symmetries, finding the ground state flux sector and understanding the QSL properties can still be challenging [59,60].
+href="#ref-eschmann2019thermodynamics" role="doc-biblioref">61,62].paragraph about amorphous lattices
In Chapter 4 I will introduce a soluble chiral amorphous quantum spin liquid by extending the Kitaev honeycomb model to random lattices with @@ -640,6 +650,7 @@ localisation.
In Chapter 3 I introduce the Long Range Falikov-Kimball Model in greater detail. I will present results that. Chapter 4 focusses on the Amorphous Kitaev Model.
+Bibliography
The Kitaev Honeycomb Model
+papers Jos on dynamics +https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.115127
intro - strong spin orbit coupling leads to anisotropic spin exchange (as opposed to isotropic exchange like the Heisenberg model) - geometrical frustration leads to QSL ground state @@ -360,6 +364,7 @@ Chern number
Phase Diagram
Bibliography
Bibliography
Bibliography
Bibliography
Takeaway: The Extended Hilbert Space decomposes into a direct product of Flux Sectors, four Topological Sectors and a set of gauge symmetries.
@@ -735,7 +737,7 @@ reduces to a determinant of the Q matrix and the fermion parity, see [2]. The only difference from the +role="doc-biblioref">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 @@ -761,7 +763,7 @@ determined by fermionic occupation numbers \(n_i\). As discussed in [2], 1], \(\hat{\pi}\) is gauge invariant in the sense that \([\hat{\pi}, D_i] = 0\).This implies that \(det(Q^u) \prod -i @@ -770,7 +772,7 @@ invariant models this quantity which can be related to the parity of the number of vortex pairs in the system [3].
+role="doc-biblioref">2].All these factors take values \(\pm
1\) so \(\mathcal{P}_0\) is 0 or
1 for a particular state. Since
More general arguments [4,3,5] imply that 4] 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
@@ -894,7 +896,7 @@ definition, the vortex free sector.
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 @@ -976,7 +978,7 @@ 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 lattices [6] and is +href="#ref-lieb_flux_1994" role="doc-biblioref">5] 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
@@ -1031,8 +1033,8 @@ 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-Chua2011" role="doc-biblioref">6–13].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 @@ -1040,7 +1042,7 @@ other by inverting the imaginary \(\phi\) fluxes [8].
+role="doc-biblioref">7].Phases of the Kitaev Model
discuss the Abelian A phase / toric code phase / anisotropic phase
@@ -1167,23 +1169,23 @@ 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 [14]." />
However, in the non-Abelian phase we have to wrangle with monodromy [4,3,5]. Monodromy is the behaviour of +role="doc-biblioref">4]. 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 @@ -1192,9 +1194,9 @@ ground state. This means that we end up with only three degenerate ground states in the non-Abelian phase \((+1, +1), (+1, -1), (-1, +1)\) [3,2,16]. Concretely, this is because the +role="doc-biblioref">15]. 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 @@ -1205,24 +1207,17 @@ proposals to use this ground state degeneracy to implement both passively fault tolerant and actively stabilised quantum computations [1,17,16,18].
+role="doc-biblioref">17,kitaevFaulttolerantQuantumComputation2003?]. +Bibliography
Bibliography
Expand on definition here
Discuss link between Chern number and Anyonic Statistics
+Bibliography
Bibliography
- Interacting Quantum Many Body Systems -
- Mott Insulators and The Hubbard Model -
- Spin Liquids -
- Exactly Solvable Models +
- Mott Insulators +
- Quantum Spin Liquids
- Outline
- The Model diff --git a/assets/thesis/intro_chapter/Venn_diagram.svg b/assets/thesis/intro_chapter/Venn_diagram.svg index 65e4d15..e59ee6a 100644 --- a/assets/thesis/intro_chapter/Venn_diagram.svg +++ b/assets/thesis/intro_chapter/Venn_diagram.svg @@ -3,10 +3,10 @@ + inkscape:current-layer="layer3" + inkscape:document-units="pt" />