diff --git a/_sass/figures.scss b/_sass/figures.scss deleted file mode 100644 index ae39bc2..0000000 --- a/_sass/figures.scss +++ /dev/null @@ -1,14 +0,0 @@ -figure { - display: flex; - flex-direction: column; - align-items: center; -} -figure img { - max-width: 900px; - width: 80%; - margin-bottom: 2em; -} -figcaption { - aria-hidden: true; - max-width: 700px; -} \ No newline at end of file diff --git a/_sass/main.scss b/_sass/main.scss index 441b17f..323ce25 100644 --- a/_sass/main.scss +++ b/_sass/main.scss @@ -4,7 +4,7 @@ @import "header"; @import "article"; @import "cv"; -@import "figures"; +@import "thesis"; * { box-sizing: border-box; @@ -66,29 +66,6 @@ img { margin-bottom: 1em; } -// For the thesis table of contents, should probably put this in a container -li { - margin-bottom: 0.2em; -} - -main > ul > li { - margin-top: 1em; -} - -main > ul > ul > li { - margin-top: 0.5em; -} - -div.csl-entry { - margin-bottom: 0.5em; -} -// div.csl-entry a { -// text-decoration: none; -// } -div.csl-entry div { - display: inline; -} - @media only screen and (max-width: $horizontal_breakpoint), only screen and (max-height: $vertical_breakpoint) diff --git a/_sass/thesis.scss b/_sass/thesis.scss new file mode 100644 index 0000000..a77c544 --- /dev/null +++ b/_sass/thesis.scss @@ -0,0 +1,48 @@ + +// Make figures looks nice +figure { + display: flex; + flex-direction: column; + align-items: center; +} +figure img { + max-width: 900px; + width: 100%; + margin-bottom: 2em; +} +figcaption { + aria-hidden: true; + max-width: 700px; +} + +// For the table of contents, should probably put this in a container + +// remove underline from toc links +nav a { + text-decoration: none; +} + +// modify the spacing of the various levels +li { + margin-bottom: 0.2em; +} + +main > ul > li { + margin-top: 1em; +} + +main > ul > ul > li { + margin-top: 0.5em; +} + +// Mess with the formatting of the citations +div.csl-entry { + margin-bottom: 0.5em; +} +// div.csl-entry a { +// text-decoration: none; +// } + +div.csl-entry div { + display: inline; +} \ No newline at end of file diff --git a/_thesis/2.1.2_AMK_Intro.html b/_thesis/2.1.2_AMK_Intro.html index 826103c..340cfae 100644 --- a/_thesis/2.1.2_AMK_Intro.html +++ b/_thesis/2.1.2_AMK_Intro.html @@ -205,8 +205,7 @@ image:
-

Properties of the Gauge -Field

+

Gauge Fields

The bond operators \(u_{ij}\) are useful because they label a bond sector \(\mathcal{\tilde{L}}_u\) in which we can @@ -513,9 +515,9 @@ take each set of \(\prod_{i \in \{i\}} D_j\) operators and gives us the complement of that set. I said earlier that \(C\) is the identity in the physical subspace and we will shortly see why.

-

W use the complement operator to rewrite the projector as a sum over -half the subsets \(\{\}\) let’s call -that \(\Lambda\). The complement +

We use the complement operator to rewrite the projector as a sum over +half the subsets of \(\{i\}\) let’s +call that \(\Lambda\). The complement operator deals with the other half

\[ \mathcal{P} = \left( \frac{1}{2^{2N-1}} \sum_{\Lambda} \prod_{i\in\{i\}} D_i\right) @@ -538,7 +540,7 @@ reduces to a determinant of the Q matrix and the fermion parity, see1 . The only difference from the +role="doc-biblioref">1. The only difference from the honeycomb case is that we cannot explicitely 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 @@ -551,19 +553,53 @@ decompositionapp:cycle_decomp?.

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

where \(p_x\;p_y\;p_z = \pm 1\) are -lattice structure factors. \(Q^u\) is -the determinant of the matrix mentioned earlier that maps \(det(Q^u)\) +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. \(\prod --i \; u_{ij}\) depend on the lattice and the particular vortex -sector. \(\hat{\pi} = \prod{i}^{N} (1 - +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 \(n_i\).

+class="math inline">\(n_i\). As discussed in +1 is \(\hat{\pi}\) 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 guage invariant quantity. In translation +invariant models this quantity which can be related to the parity of the +number of vortex pairs in the system2. However it is not so simple to +evaluate in the amorphous case.

+

More general arguments3,4 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 on of the four +topological sectors. This forces that sector that contain a fermion and +hence gives the model a three-fold degerenate ground state. In the +Abelian phase this doesn’t happen and we get a fourfold degerate ground +state. Whether this analysis generalises to the amorphous case in +unclear.

+

An alternate way to view this is to consider the adiabatic insertion +of the fluxes \(\Phi_{x,y}\) as the +operations that undo vortex transport around the lattice. In this +picture the three fold degeneracy occurs because transporting a vortex +around both the major and minor axes of the torus +changes its fusion channel such that the two vortices fuse into a +fermion excition rather than the vacuum.

All these factors take values \(\pm 1\) so \(\mathcal{P}_0\) is 0 or 1 for a particular state. Since basis we would need to include the full symmetrisation over the gauge fields. However this was not necessary for any of the results that will be presented here.

-

Open boundary conditions

-

Care must be taken in the definition of open boundary conditions. -Simply removing bonds from the lattice leaves behind unpaired \(b^\alpha\) operators that need to be paired -in some way to arrive at fermionic modes. In order to fix a pairing we -always start from a lattice defined on the torus and generate a lattice -with open boundary conditions by defining the bond coupling \(J^{\alpha}_{ij} = 0\) for sites joined by -bonds \((i,j)\) that we want to remove. -This creates fermionic zero modes \(u_{ij}\) associated with these cut bonds -which we set to 1 when calculating the projector.

-

Alternatively, since all the fermionic zero modes are degenerate -anyway, an arbitrary pairing of the unpaired \(b^\alpha\) operators could be performed. -Is is possible that a lattice constructed and coloured like this -would have unequal numbers of \(b^x\) -\(b^y\) and \(b^z\) operators?

-

The Ground State Vortex -Sector

+

The Ground State

+

As we have shown that the Hamiltonian is gauge invariant, only the +flux sector and the two topological fluxes affect the spectrum of the +Hamiltonian. Thus we can label many body ground state by a combination +of flux sector and fermionic occupation numbers.

+

By studying the projector we saw that the fermionic occupation +numbers of the ground state will always be either \(n_m = 0\) or \(n_0 = 1, n_{m>1} = 0\) because the +projector really just enforces vortex and fermion parity.

+

I refer to the flux sector that contains the ground state as the +ground state flux sector. Recall that we call the excitations of the +fluxes away from the ground ground state configuration +vortices, so that the ground state flux sector is the +vortex free sector by definition.

On the Honeycomb, Lieb’s theorem implies that the the ground state -corresponds to the state where all \(u_jk = +corresponds to the state where all \(u_{jk} = 1\) implying that the flux free sector is the ground state sector2.

+href="#ref-lieb_flux_1994" role="doc-biblioref">5.

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 @@ -713,7 +743,7 @@ 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 lattices2 and +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 I call a vortex.

@@ -723,8 +753,36 @@ effects. In order to rigorously test it we would like to directly enumerate the \(2^N\) vortex sectors for a smaller lattice and check that the lowest state found is the vortex sector predicted by ???.

-

To do this we tile an amorphous lattice onto a repeating \(NxN\) grid. The use of a fourier series +

To do this we tile use an amorphous lattice as the unit cell of a +periodic \(N\times N\) system. Bonds +that originally crossed the periodic boundaries now connect adjacent +unit cells. Using Bloch’s theorem the problem then essnetially reduces +back to the single amorphous unit cell but now the edges that cross the +periodic boundaries pick up a phase dependent on the crystal momentum +\(\vex{q} = (q_x, q_y)\) and the +lattice vector of the bond \(\vec{x} = (+1, 0, +-1, +1, 0, -1)\). Assigning these lattice vectors to each bond is +also a very conveninent way to store and plot toroidal graphs.

+

This can then be solved using Bloch’s theorem. For a given crystal +momentum \(\textbf{q} \in [0,2\pi)^2\), +we are left with a Bloch Hamiltonian, which is identical to the original +Hamiltonian aside from an extra phase on edges that cross the periodic +boundaries in the \(x\) and \(y\) directions, \[\begin{aligned} + M_{jk}(\textbf{q}) = \frac{i}{2} J^{\alpha} u_{jk} e^{i +q_{jk}},\end{aligned}\] where \(q_{jk} += q_x\) for a bond that crosses the \(x\)-periodic boundary in the positive +direction, with the analogous definition for \(y\)-crossing bonds. We also have \(q_{jk} = -q_{kj}\). Finally \(q_{jk} = 0\) if the edge does not cross any +boundaries at all – in essence we are imposing twisted boundary +conditions on our system. The total energy of the tiled system can be +calculated by summing the energy of \(M( +\textbf{q})\) for every value of \(\textbf{q}\). The use of a fourier series then allows us to compute the diagonalisation with a penalty only linear in the number of tiles used compared to diagonalising a single lattice. With this technique the finite size effects related to the @@ -736,16 +794,16 @@ class="math inline">\(\phi_0\) correctly predicts the ground state for hundreds of thousands of lattices with upto 20 plaquettes. For larger lattices we verified that random perturbations around the predicted ground state never yield a lower energy state.

-

Chiral Symmetry

+

Chiral Symmetry

In the discussion above we see that the ground state has a twofold chiral degeneracy that comes about because the global sign of the odd plaquettes does not matter.

This happens because by adding odd plaquettes we have broken the time reversal symmetry of the original model3–6–10.

+role="doc-biblioref">13.

Similar to the behaviour of the original Kitaev model in response to a magnetic field, we get two degenerate ground states of different handedness. Practicaly speaking, one ground state is related to the @@ -753,9 +811,16 @@ other by inverting the imaginary \(\phi\) fluxes4.

-

Topology, chirality and edge -modes

+role="doc-biblioref">7.

+

Phases of the Kitaev Model

+

discuss the abelian A phase / toric code phase / anisotropic +phase

+

the isotropic gapless phase of the standard model

+

The isotropic gapped phase with the addition of a magnetic field

+

What’s so great about two +dimensions?

+

Topology, chirality and edge +modes

Most thermodynamic and quantum phases studied can be characterised by a local order parameter. That is, a function or operator that only requires knowledge about some fixed sized patch of the system that does @@ -772,7 +837,7 @@ breaking.

looked at it defined on a graph that is embedded either into the plane or onto the torus. The extension to surfaces like the torus but with more than one handle is relatively easy.

-

Anyonic Statistics

+

Anyonic Statistics

In dimensions greater than two, the quantum state of a system must pick up a factor of \(-1\) or \(+1\) if two identical particles are @@ -884,23 +949,23 @@ class="math inline">\((+1, +1), (+1, -1), (-1, +1), (-1,

+alt="Figure 8: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the donut/torus or through the filling. If they made donuts that had both a jam filling and a hole this analogy would be a lot easier to make14." /> +role="doc-biblioref">14.

However in the non-Abelian phase we have to wrangle with monodromy12,3,13. Monodromy is behaviour of +role="doc-biblioref">4. Monodromy is 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 anhilating to the @@ -910,7 +975,7 @@ the non-Abelian phase \((+1, +1), (+1, -1), (-1, +1)\)14. The way that this shows up +role="doc-biblioref">15. The way that this shows up concretly is that 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 which forces means we have to introduce a @@ -928,10 +993,10 @@ passively fault tolerant and actively stabilised quantum computations [15;16;16; +role="doc-biblioref">17; hastingsDynamicallyGeneratedLogical2021].

@@ -943,8 +1008,34 @@ href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of the Kitaev honeycomb model. Phys. Rev. B 84, 165414 (2011).
+
+
2.
Yao, +H., Zhang, S.-C. & Kivelson, S. A. Algebraic +Spin Liquid in an Exactly Solvable Spin +Model. Phys. Rev. Lett. 102, 217202 +(2009).
+
+
+
3.
Chung, S. B. & Stone, M. Explicit monodromy of +Moore wavefunctions on a torus. J. Phys. A: Math. +Theor. 40, 4923–4947 (2007).
+
+
+
4.
Oshikawa, M., Kim, Y. B., Shtengel, K., Nayak, +C. & Tewari, S. Topological degeneracy +of non-Abelian states for dummies. Annals of +Physics 322, 1477–1498 (2007).
+
-
2.
5.
Lieb, E. H. Flux Phase of the Half-Filled Band. @@ -952,7 +1043,7 @@ href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux (1994).
-
3.
6.
Chua, V., Yao, H. & Fiete, G. A. Exact chiral spin liquid with stable spin Fermi surface on the kagome @@ -961,21 +1052,21 @@ lattice. Phys. Rev. B 83, 180412
-
4.
Yao, +
7.
Yao, H. & Kivelson, S. A. An exact chiral spin liquid with non-Abelian anyons. Phys. Rev. Lett. 99, 247203 (2007).
-
5.
8.
Chua, V. & Fiete, G. A. Exactly solvable topological chiral spin liquid with random exchange. Phys. Rev. B 84, 195129 (2011).
-
6.
9.
Fiete, G. A. et al. Topological insulators and quantum spin liquids. Physica E: Low-dimensional @@ -983,20 +1074,20 @@ Systems and Nanostructures 44, 845–859 (2012).
-
7.
10.
Natori, W. M. H., Andrade, E. C., Miranda, E. & Pereira, R. G. Chiral spin-orbital liquids with nodal lines. Phys. Rev. Lett. 117, 017204 (2016).
-
8.
Wu, +
11.
Wu, C., Arovas, D. & Hung, H.-H. \(\Gamma\)-matrix generalization of the Kitaev model. Physical Review B 79, 134427 (2009).
-
9.
12.
Peri, V. et al. Non-Abelian chiral spin liquid on a simple non-Archimedean lattice. @@ -1004,7 +1095,7 @@ chiral spin liquid on a simple non-Archimedean lattice.
-
10.
13.
Wang, H. & Principi, A. Majorana edge and corner states in square and kagome quantum spin-\(^{3}\fracslash_2\) liquids.
-
11.
14.
Parker, M. Why does this balloon have -1 holes?
-
-
12.
Chung, S. B. & Stone, M. Explicit monodromy of -Moore wavefunctions on a torus. J. Phys. A: Math. -Theor. 40, 4923–4947 (2007).
-
-
-
13.
Oshikawa, M., Kim, Y. B., Shtengel, K., Nayak, -C. & Tewari, S. Topological degeneracy -of non-Abelian states for dummies. Annals of -Physics 322, 1477–1498 (2007).
-
-
14.
15.
Chung, S. B., Yao, H., Hughes, T. L. & Kim, E.-A. Topological quantum phase transition in an exactly solvable model of a chiral spin @@ -1045,7 +1119,7 @@ liquid at finite temperature. Phys. Rev. B
-
15.
16.
Kitaev, A. Yu. Fault-tolerant quantum computation by anyons. Annals of Physics @@ -1053,7 +1127,7 @@ quantum computation by anyons. Annals of Physics
-
16.
17.
Poulin, D. Stabilizer Formalism for Operator Quantum Error diff --git a/_thesis/2.1_AMK_Intro.html b/_thesis/2.1_AMK_Intro.html index 8427b02..059f880 100644 --- a/_thesis/2.1_AMK_Intro.html +++ b/_thesis/2.1_AMK_Intro.html @@ -209,39 +209,29 @@ image: -
  • An in-depth look at the -Kitaev Model +
  • The Kitaev +Model
  • The Majorana Hamiltonian
    • +id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian + @@ -278,10 +268,6 @@ href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">4. Its solubility comes about because the model has extensively many conserved degrees of freedom that mediate the interactions between quantum degrees of freedom.

    -

    Amorphous Systems

    -

    Insert discussion of why a generalisation to the amorphous -case is intersting

    -

    Chapter outline

    In this chapter I will discuss the physics of the Kitaev Model on amorphous lattices.

    I’ll start by discussing the physics of the Kitaev model in much more @@ -320,13 +306,15 @@ addtional small Heisenberg term \[H_{KH} = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} + \sigma_j\sigma_k\]

    -

    An in-depth look at the -Kitaev Model

    -

    Commutation relations

    +

    Amorphous Systems

    +

    Insert discussion of why a generalisation to the amorphous +case is intersting

    +

    The Kitaev Model

    +

    Commutation relations

    Before diving into the Hamiltonian of the Kitaev Model, here is a quick refresher of the key commutation relations of spins, fermions and Majoranas.

    -

    Spins

    +

    Spins

    Skip this is you’re super familiar with the algebra of the Pauli martrices. Scalars like \(\delta_{ij}\) should be understood to be multiplied by an implicit identity \[\sigma^\alpha \sigma^\beta \sigma^\gamma = i \epsilon^{\alpha\beta\gamma}\] and \[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] = 0\]

    -

    Fermions and Majoranas

    +

    Fermions and Majoranas

    The fermionic creation and anhilation operators are defined by the canonical anticommutation relations \[\begin{aligned} @@ -387,7 +375,7 @@ class="math inline">\(c_i\). The property that must be preserved however is that the Majoranas still anticommute:

    \[ \{c_i, c_j\} = 2\delta_{ij}\]

    -

    The Hamiltonian

    +

    The Hamiltonian

    To get down to brass tacks, the Kitaev Honeycomb model is a model of interacting spin\(-1/2\)s on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a @@ -497,9 +485,9 @@ of a plaqutte operator away from the ground state as Hilbert space into a set of ‘vortex sectors’ labelled by that particular flux configuration \(\phi_i = \pm 1,\pm i\).

    -

    From Spins to Majorana -operators

    -

    For a single spin

    +

    From Spins to Majorana +operators

    +

    For a single spin

    Let’s start by considering just one site and its \(\sigma^x, \sigma^y\) and \(\sigma^z\) operators which live in a two @@ -560,7 +548,7 @@ alt="Figure 4: " />

    -

    For multiple spins

    +

    For multiple spins

    This construction generalises easily to the case of multiple spins: we get a set of 4 Majoranas \(b^x_j,\; b^y_j,\;b^z_j,\; c_j\) and a \(D_j = @@ -612,8 +600,8 @@ degree of degeneracy.

    In summary Majorana bond operators \(u_{ij}\) are an emergent, classical, \(\mathbb{Z_2}\) gauge field!

    -

    Partitioning -the Hilbert Space into Bond sectors

    +

    Partitioning +the Hilbert Space into Bond sectors

    Similar to the story with the plaquette operators from the spin language, we can break the Hilbert space \(\mathcal{L}\) up into sectors labelled by @@ -713,8 +701,8 @@ can take half the absolute value of the whole set to recover \(\sum_m \epsilon_m\) easily.

    The Majorana Hamiltonian is quadratic within a Bond Sector.

    -

    Mapping -back from Bond Sectors to the Physical Subspace

    +

    Mapping +back from Bond Sectors to the Physical Subspace

    At this point, given a particular bond configuration \(u_{ij} = \pm 1\) we are able to construct a quadratic Hamiltonian \(\tilde{H}_u\) @@ -771,6 +759,25 @@ class="math inline">\(u_{ij}\), these will turn out to just be the plaquette operators.

    The Bond Sectors overlap with the physical subspace but are not contained within it.

    +

    Open boundary conditions

    +

    Care must be taken in the definition of open boundary conditions. +Simply removing bonds from the lattice leaves behind unpaired \(b^\alpha\) operators that need to be paired +in some way to arrive at fermionic modes. In order to fix a pairing we +always start from a lattice defined on the torus and generate a lattice +with open boundary conditions by defining the bond coupling \(J^{\alpha}_{ij} = 0\) for sites joined by +bonds \((i,j)\) that we want to remove. +This creates fermionic zero modes \(u_{ij}\) associated with these cut bonds +which we set to 1 when calculating the projector.

    +

    Alternatively, since all the fermionic zero modes are degenerate +anyway, an arbitrary pairing of the unpaired \(b^\alpha\) operators could be performed. +Is is possible that a lattice constructed and coloured like this +would have unequal numbers of \(b^x\) +\(b^y\) and \(b^z\) operators?

  • Coloring the Bonds
  • Mapping between flux sectors and bond sectors
    • +
  • Chern +Markers
    • @@ -289,10 +297,49 @@ edges.

    Graph Representation

    -

    We represent the graph structure with an ordered list of edges \((i,j)\) so we can represent both directed -and undirected graphs which is useful for defining the sign of bond -operators \(u_{ij} = - u_{ji}\).

    +

    There are three keys pieces of information that we use to represent +amorphous lattices.

    +

    Most of the graph connectivity is encoded by an ordered list of edges +\((i,j)\). These are ordered so that we +can represent both directed and undirected graphs which is useful for +defining the sign of bond operators \(u_{ij} = +- u_{ji}\).

    +

    Information about the embedding of the lattice onto the torus is +encoded into a point on the unit square associated with each vertex. The +torus is unwrapped onto the square by defining an arbitary pair of cuts +along the major and minor axes which for simplicity we take to be the +lines \(x = 0\) and \(y = 0\). We can wrap the unit square back +up into a torus by identifying the lines \(x = +0\) with \(x = 1\) and \(y = 0\) with \(y += 1\).

    +

    Finally, we need a way to encode the topology of the graph. We need +this because given simply an edge \((i, +j)\) we do not know how the edge gets from vertex i to vertex j. +It could do so by taking the shortest path but it could also ‘go the +long way around’ by crossing one of the cuts. To encode this information +we store an additional vector \(\vec{r}\) associated with each edge. \(r_i^x = 0\) means that edge i does not +cross the x while \(r_i^x = +1\) (\(-1\)) means it crossed the cut in a +positive (negative) sense.

    +

    This description of the lattice has a very nice relationship to +Bloch’s theorem. When you apply Bloch’s theorem to a periodic lattice +you essentially wrap the unit cell onto a torus. Variations that happen +at longer length scales than the size of the unit cell are captured by +the crystal momentum which inserts a phase factor \(e^{i \vec{q}\cdot\vec{r}}\) onto bonds that +cross to adjacent unit cells. The vector \(\vec{r}\) is exactly what we use to encode +the topology of our lattices.

    +
    + + +

    Coloring the Bonds

    The Kitaev model requires that each edge in the lattice be assigned a label \(x\), 12 . For which there is an \(\mathcal{O}(n^2)\) algorithm robertson1996efficiently . However it is not clear whether this extends to cubic, toroidal bridgeless graphs.

    -

    4-face-colourablity -implies 3-edge-colourability

    +

    Four-colourings and +three-colourings

    The proof of that 4-face-colourablity implies 3-edge-colourability can be sketched out quite easily: 1. Assume the faces of G can be 4-coloured with labels (0,1,2,3) 2. Label each edge of G according to @@ -443,6 +489,19 @@ valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted.

    +

    Does it matter which +colouring we choose?

    +

    In the isotropic case \(J^\alpha = +1\) it is easy to show that it can’t possibly make a difference. +As the choice of how we define the four Majoranas at a site is arbitrary +we can define a local operator that tranforms the colouring of any +particular site to another permutation. The operators commute with the +Hamiltonian and by composing such operators we can tranform the +Hamiltonian generated by one colouring into that generated by +another.

    +

    We can’t do this in the anisotropic case however, and it remains an +open question whether particular physical properties could arise by +engineering the colouring in this phase.

    Mapping between flux sectors and bond sectors

    Constructing the Majorana representation of the model requires the @@ -501,689 +560,19 @@ flux has will remain because the starting and target flux sectors differed by an odd number of fluxes.

    -

    Amorphous materials are glassy condensed matter systems characterised -by short-range constraints in the absence of long-range crystalline -order as first studied in amorphous semiconductors15,16. -In general, the bonds of a whole range of covalent compounds enforce -local constraints around each ion, e.g.~a fixed coordination number -\(z\), which has enabled the prediction -of energy gaps even in lattices without translational symmetry17,18, the most famous example being -amorphous Ge and Si with \(z=4\)19,20. Recently, following the -discovery of topological insulators (TIs) it has been shown that similar -phases can exist in amorphous systems characterized by protected edge -states and topological bulk invariants2,3,21–25. However, research on -electronic systems has been mostly focused on non-interacting systems -with a few notable exceptions for understanding the occurrence of -superconductivity26–28 in amorphous materials and -recently the effect of strong repulsion in amorphous TIs29.

    -

    Magnetic phases in amorphous systems have been investigated since the -1960s, mostly through the adaptation of theoretical tools developed for -disordered systems and numerical methods~. Research focused on classical -Heisenberg and Ising models which have been shown to account for -observed behavior of ferromagnetism, disordered antiferromagnetism and -widely observed spin glass behaviour~. However, the role of -spin-anisotropic interactions and quantum effects has not been -addressed. Similarly, it is an open question whether magnetic -frustration in amorphous quantum magnets can give rise to long-range -entangled quantum spin liquid (QSL) phases.

    -

    %Broad constraints to the possible phases hosted by Heisenberg -amorphous magnets were provided by the phenomenological theory developed -by Andreev and Marchenko30–32. The phases in this theory are -described by a set of macroscopic magnetic vectors that transform -according to the irreducible representations of the group of spatial -symmetries of the system30. Amorphous magnets are treated, -on average, as homogeneous and isotropic, being thus symmetric under -three-dimensional rotations and spatial inversion31. Only three types of phases are -consistent to this group of symmetries, corresponding to ferromagnets, -disordered antiferromagnets, or spin glasses31,32.

    -

    Two intentional simplifications of Andreev’s and Marchenko’s theory -were the neglect of spin-orbit coupling induced anisotropies and the -effects arising from the local structure of amorphous lattices. It is -then expected that their theory is invalid for amorphous compounds -generated from crystalline magnets with strong spin-orbit coupling with -tight geometrical arrangements. Several instances of these magnets were -synthesized in the last decade, among which we highlight the Kitaev -materials33–37. It was -suggested (and later observed) that heavy-ion Mott insulators formed by -edge-sharing octahedra could be good platforms for the celebrated Kitaev -model on the honeycomb lattice33, an exactly solvable model -whose ground state is a quantum spin liquid (QSL)38–41 -characterized by a static \(\mathbb -Z_2\) gauge field and Majorana fermion excitations42. The model displays -bond-dependent Ising-like exchanges that give rise to local symmetries, -which are essential to its mapping onto a free fermion problem43,44. Such -a mapping is rigorously extendable to any three-coordinated graph in two -or three dimensions satisfying a simple geometrical condition45–48. Thus, it -reasonable to suppose that the Kitaev model is also analytically -treatable on certain amorphous lattices, therefore becoming a realistic -starting point to study the overlooked possibility of QSLs in amorphous -magnets.

    -In this letter, we study Kitaev spin liquids (KSLs) stabilized by the -\(S=1/2\) Kitaev model on coordination -number \(z=3\) random networks -generated via Voronoi tessellation . On these lattices, the KSLs -generically break time-reversal symmetry (TRS), as expected for any -Majorana QSL in graphs containing odd-sided plaquettes . An extensive -numerical study showed that the \(\mathbb -Z_2\) gauge fluxes on the ground state can be described by a -conjecture consistent with Lieb’s theorem . In contrast to the honeycomb -case, the amorphous KSLs are gapless only along certain critical lines. -These manifolds separate two gapped KSLs that are topologically -differentiated by a local Chern number \(\nu\) in analogy with the KSLs on the -decorated honeycomb lattice . The \(\nu=0\) phase is the amorphous analogue of -the abelian toric-code QSL , whereas the \(\nu=\pm1\) KSLs is a non-Abelian chiral -spin liquid (CSL). We study two specific features of the latter liquid: -topologically protected edge states and a thermal-induced Anderson -transition to a thermal metal phase . -

    % The Kitaev spin liquids are classified by their Majorana fermion -dispersion and topological properties. On the honeycomb lattice, tuning -the exchange couplings \(J^\alpha\) can -change the ground state from a gapped QSL with Abelian anyonic -excitations (e.g., when \(J^z\gg -J^x,J^y\)) or gapless (e.g., when \(J^z=J^x=J^y\)). In the latter case, -breaking time reversal symmetry (TRS) opens a gap that signals the onset -of a chiral spin liquid (CSL) phase supporting non-Abelian excitations -and protected edge modes. On the honeycomb lattice, CSLs are only -obtained by perturbing a Hamiltonian with, for example, magnetic fields -or Dzyaloshinskii-Moriya exchanges . CSLs on the pure Kitaev model can -be obtained on \(z=3\) lattices -containing odd-sided plaquettes, for which any Majorana QSL displays -spontaneous TRS breaking , as confirmed on decorated honeycomb and -non-Archimedean lattices.

    -{} We start with a brief review of the Kitaev model on the honeycomb -lattice . Here, a spin-1/2 is placed on every vertex and each bond is -labelled by an index \(\alpha \in \{ x, y, -z\}\). The bonds are arranged such that each vertex connects to -exactly one bond of each type. The Hamiltonian is given by \[\begin{equation} - \label{eqn:kitham} - \mathcal{H} = - \sum_{\langle j,k\rangle_\alpha} -J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha}, -\end{equation}\] where \(\sigma^\alpha_j\) is a Pauli matrix acting -on site \(j\), (j,k_) is a pair of -nearest-neighbour indices connected by an \(\alpha\)-bond with exchange coupling \(J^\alpha\). For each plaquette of the -lattice, we can define the conserved operator $ W_p = _j^{}_k^{}$, where -the product runs clockwise over the bonds around the plaquette. This -provides an extensive number of conserved plaquettes that allow us to -split the Hilbert space in terms of the eigenvalues of \(W_p\). -The KSL is uncovered by transforming eqn.~\(\ref{eqn:kitham}\) to a four-Majorana -representation of the spin operators, \(\sigma_i^\alpha = i b_i^\alpha c_i\) , -where the Hamiltonian takes the form \[\begin{equation}\label{eqn:majorana_hamiltonian} - \mathcal{H} = \frac{i}{4}\sum_{j,k}A_{jk}^{(\alpha)}c_jc_k. -\end{equation}\] Here, \(A_{jk}^{(\alpha)}=2J^{\alpha}u_{jk}\) with -\(\hat u_{jk} = -ib_j^{\alpha}b_k^{\alpha}\) being conserved \(\mathbb Z_2\) bond operators. Once the -\(\hat u_{jk}\) eigenvalues are fixed, -the Kitaev model becomes equivalent to a fermionic problem that can be -diagonalized with standard methods . -The Kitaev Hamiltonian remains exactly solvable on any graph in which no -site connects to more than one bond of the same type . Thus, we are -restricted to lattices in which every vertex has coordination number -\(z \leq 3\). Here, such graphs are -generated with Voronoi tessellation . A set of points are sampled -uniformly from the unit square and cells are generated as the region of -space closer to a given point than any other. The lattice is given by -the boundaries between cells with edges at the interface of two cells -and vertices at the point where three edges meet. Periodic boundary -conditions are imposed by tiling the initial set of points and then -connecting corresponding edges that cross the unit square boundaries - -see for technical details. One example of such an amorphous lattice is -shown in~(a). -Once a random network has been generated, the bonds types must be -assigned in a way that is consistent with our condition, which we refer -to as a . The problem of finding such a colouring was shown to be -equivalent to the classical problem of four-colouring the faces, which -is always solvable in planar graphs~. On the torus, a face colouring can -require up to seven colours , and so not all graphs can be assumed to be -3-edge colourable. However, such exceptions are rare – every graph -generated in this study admitted multiple distinct 3-edge colourings. -The problem of finding a colouring for a given graph can be reduced to a -Boolean satisfiability problem , which we then solve using the -open-source solver ~. -

    Once the three-edge colouring has been found, the Kitaev Hamiltonian -is mapped onto eqn.~\(\ref{eqn:majorana_hamiltonian}\), which -corresponds to the spin fractionalization in terms of a static \(\mathbb Z_2\) gauge fields and \(c\) matter as indicated in ~(b) . Strictly -speaking, the Majorana system is equivalent to the original spin system -after applying a projector operator , whose form is presented in . -Despite this caveat, one can still use eqn.~\(\ref{eqn:majorana_hamiltonian}\) to -evaluate the expectation values of operators conserving \(\hat u_{jk}\) in the thermodynamic limit . -This type of operator is exemplified by the Hamiltonian itself, for -which the ground state energy of a fixed sector is the sum of the -negative eigenvalues of \(iA/4\) in -eqn.~\(\ref{eqn:majorana_hamiltonian}\), and whose -excitations are extracted from the positive eigenvalues of the same -matrix.

    -{} Let us now consider the conserved operators $ W_p = _j^{}_k^{}$ on -amorphous lattices. When represented in the Majorana Hilbert space, -these operators correspond to ordered products of \(\hat u_{jk}\), and their fixed eigenvalues -are written as \[\begin{equation} -\label{eqn:flux_definition} - \phi_p = \prod_{(j,k) \in \partial p} (-iu_{jk}), -\end{equation}\] where the pairs \(j,k\) are crossed around the border \(\partial p\) of the plaquette on the -orientation. In periodic boundaries there is an additional pair of -global \(\mathbb{Z}_2\) fluxes \(\Phi_x\) and \(\Phi_y\), which are calculated along an -arbitrary closed path that wraps the torus in the \(x\) and \(y\) directions respectively. The energy -difference between distinct flux sectors decays exponentially with -system size, so that the ground state of any flux sector in the -thermodynamic limit displays a fourfold topological degeneracy . -We now need to determine the ground state flux sectors. First, let us -recall that Majorana QSLs emerging on graphs containing odd-sided -plaquette undergo a spontaneous TRS breaking . Therefore, there will be -always a twofold ground state degeneracy due to time-reversal, in which -one ground state is related to the other by inversion of imaginary \(\phi_p\) fluxes . An insight pointing to -the ground state sectors come from the model on the honeycomb lattice, -for which a theorem proved by Lieb sets that the ground state sector to -be \(\phi_p=+1\), \(\forall p\) . Although Lieb’s theorem is -not extendable to amorphous lattices, it is suggested the ground state -energy for a sufficiently large system is minimised by setting \[\begin{align} \label{eqn:gnd_flux} - \phi_p^{\textup{g.s.}} = -(\pm i)^{n_{\textup{sides}}}, -\end{align}\] where \(n_{\textup{sides}}\) is the number of edges -that form \(p\) and the global choice -of the sign of \(i\) gives each of the -two TRS-degenerate ground state flux sectors. Such a conjecture is -consistent with Lieb’s theorem on regular lattices and is supported by -numerical evidence as detailed in . Once we have identified the ground -state, any other sector can be characterized by the configuration of -vortices, i.e. by the plaquettes whose flux is flipped with respect to -\(\left\{\phi_p^{\textup{g.s.}}\right\}\). -{} We numerically found that the amorphous KSLs are generally gapped, -except along the critical lines displayed in \(\ref{fig:example_lattice}\)(c). The QSLs -separated by these lines are distinguished by a real-space analogue of -the Chern number . A similar topological number was discussed by Kitaev -on the honeycomb lattice that we shall use here with a slight -modification . For a choice of flux sector, we calculate the projector -\(P\) onto the negative energy -eigenstates of the matrix \(iA\) -defined in eqn.~\(\ref{eqn:majorana_hamiltonian}\). The local -Chern number around a point \(\bf R\) -in the bulk is given by \[\begin{align} - \nu (\bf R) = 4\pi \Im \Tr_{\textup{Bulk}} - \left ( - P\theta_{R_x} P \theta_{R_y} P - \right ), -\end{align}\] where \(\theta_{R_x}\) is a step function in the -\(x\)-direction, with the step located -at \(x = R_x\), \(\theta_{R_y}\) is defined analogously. The -trace is taken over a region around \(\bf -R\) in the bulk of the material, where care must be taken not to -include any points close to the edges. Provided that the point \(\bf R\) is sufficiently far from the edges, -this quantity will be very close to quantised to the Chern number. -The local Chern marker distinguishes between an Abelian phase (A) with -\(\nu = 0\), and a non-Abelian (B) -phase characterized by \(\nu = \pm 1\). -The (A) phase is equivalent to the toric code on an amorphous system . { -Since the (A) phase displays the “topological” degeneracy described -above, I think that “topologically trivial” is not a good term to -describe it. Another thing that I think it should be considered here. -The abelian phase is expected to have 2x4 degeneracy, where the factor -of 2 comes from time-reversal. On the other hand, the non-Abelian phase -should display 2x3 degeneracy, as discussed by . Did you get any -evidence of this?} -By contrast, the (B) phase is a , the magnetic analogue of the -fractional quantum Hall state. Topologically protected edge modes are -predicted to occur in these states on periodic boundary conditions -following the bulk-boundary correspondence . The probability density of -one such edge mode is given in (a), where it is shown to be -exponentially localised to the boundary of the system. The localization -of these modes can be quantified by their inverse participation ratio -(IPR), \[\begin{equation} - \textup{IPR} = \int d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau}, -\end{equation}\] where \(L\sim\sqrt{N}\) is the characteristic -linear dimension of the amorphous lattices and \(\tau\) dimensional scaling exponent of IPR. -Finally, the CSL density of states in open boundary conditions indicates -the low-energy modes within the gap of Majorana bands in (b). { Could -you plot the dimensional scaling exponent \(\tau\) in (a)?} -

    The phase diagram of the amorphous model in \(\ref{fig:example_lattice}\)(c) displays a -reduced parameter space for the non-Abelian phase when compared to the -honeycomb model. Interestingly, similar inward deformations of the -critical lines were found on the Kitaev honeycomb model subject to -disorder by proliferating flux vortices or exchange disorder .

    -{} An Ising non-Abelian anyon is formed by Majorana zero-modes bound to -a topological defect . Interactions between anyons are modeled by -pairwise projectors whose strength absolute value decays exponentially -with the separation between the particles, and whose sign oscillates in -analogy to RKKY exchanges . Disorder can induce a finite density of -anyons whose hybridization lead to a macroscopically degenerate state -known as . One instance of this phase can be settled on the Kitaev CSL. -In this case, the topological defects correspond to the \(W_p \neq +1\) fluxes, which naturally -emerge from thermal fluctuations at nonzero temperature . -We demonstrated that the amorphous CSL undergoes the same form of -Anderson transition by studying its properties as a function of -disorder. Unfortunately, we could not perform a complete study of its -properties as a function of the temperature as it was not feasible to -evaluate an ever-present boundary condition dependent factor for random -networks. Instead, we evaluated the fermionic density of states (DOS) -and the IPR as a function of the vortex density \(\rho\) as a proxy for temperature. This -approximation is exact in the limits \(T = -0\) (corresponding to \(\rho = -0\)) and \(T \to \infty\) -(corresponding to \(\rho = 0.5\)). At -intermediate temperatures the method neglects to include the influence -of defect-defect correlations. However, such an approximation is enough -to show the onset of low-energy excitations for \(\rho \sim 10^{-2}-10^{-1}\), as displayed -on the top graphic of \(\ref{fig:DOS_Oscillations}\)(a). We -characterized these gapless excitations using the dimensional scaling -exponential \(\tau\) of the IPR on the -bottom graphic of the same figure. At small \(\rho\), the states populating the gap -possess \(\tau\approx0\), indicating -that they are localised states pinned to the defects, 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 a metallic phase. { Maybe being a -bit more quantitative about \(\tau\) -can enrich the discussion by allowing us to discuss a bit about the -multifractality of these low-energy states} -The thermal metal DOS displays a logarithmic divergence at zero energy -and characteristic oscillations at small energies. . These features were -indeed observed by the averaged density of states in the \(\rho = 0.5\) case shown in (b) for -amorphous lattice. We emphasize that the CSL studied here emerges -without an applied magnetic field as opposed to the CSL on the honeycomb -lattice studied in Ref. { I have the impression that (b) on the top is -very similar to Fig. 3 of . Maybe a more instructive figure would be the -DOS of the amorphous toric code at the infinite temperature limit. In -this case, the lack of non-Abelian anyons would be reflected by a gap on -the DOS, which would contrast nicely to the thermal metal phase} -

    % This high temperature phase of the amorphous model is known as a -thermal metal. The signature of the thermal metal phase is -characteristic oscillations in the low energy density of states, as seen -in~(b).

    -{} We have studied an extension of the Kitaev honeycomb model to -amorphous lattices with coordination number \(z= 3\). We found that it is able to support -two quantum spin liquid phases that can be distinguished using a -real-space generalisation of the Chern number. The presence of odd-sided -plaquettes on these lattices let to a spontaneous breaking of time -reversal symmetry, leading to the emergence of a chiral spin liquid -phase. Furthermore we found evidence that the amorphous system undergoes -an Anderson transition to a thermal metal phase, driven by the -proliferation of vortices with increasing temperature. The next step is -to search for an experimental realisation in amorphous Kitaev materials, -which can be created from crystalline ones using several methods . -Following the evidence for an induced chiral spin liquid phase in -crystalline Kitaev materials , it would be interesting to investigate if -a similar state is produced on its amorphous counterpart. Besides the -usual half-quantized signature on thermal Hall effect , such a CSL could -be also characterized using local probes such as spin-polarized -scanning-tunneling microscopy . The same probes would also be useful to -manipulate non-Abelian anyons , thereby implementing elementary -operations for topological quantum computation. Finally, the thermal -metal phase can be diagnosed using bulk heat transport measurements . -

    This work can be generalized in several ways. Introduction of -symmetry allowed perturbations on the model . Generalizations to -higher-spin models in random networks with different coordination -numbers

    -

    % In the present work, we have avoided the need for a rigorous Monte -Carlo study of the thermal phase transition. As a consequence, the -thermodynamic nature of the transition between the chiral QSL and -thermal metal states has not been elucidated. { insert some guff about -the Imri-Ma argument}.

    -

    { Probably one way to make this theory experimentally relevant is to -do experiments on amorphous phases of Kitaev materials. These phases can -be obtained by liquifying the material and cooling it fast. Apparently, -most of crystalline magnets can be transformed into amorphous ones -through this process. } %Metal-organic frameworks (MOFs) are a promising -candidate for realising Kitaev physics in an amorphous system. Yamada et -al. propose a realisation of the Kitaev honeycomb model in a crystalline -Ru-oxalate MOF~, and Misumi et al.~have demonstrated potential -signatures of a resonating valence bond quantum spin liquid state in -MOFs with Kagome geometry~. Amorphous MOFs can be generated by -introducing disorder into crystalline MOFs through mechanical -processes~, suggesting a natural route to realising amorphous Kitaev -physics. Assuming it is possible to realise Kitaev physics in a -crystalline MOF, it is unclear what superexchange couplings would be -retained when disorder is introduced to the lattice. Because it is -unlikely one would cleanly reproduce the exact model described in the -present work, future work should examine how robust the CSL ground state -of the amorphous Kitaev model is to additional disorder in the -Hamiltonian, for example random recoloring of the bonds, additional bond -forming and breaking, and disorder in coupling strengths.

    -

    % Produces the bibliography via BibTeX. %

    -

    A random pointset is used to partition space into polyhedral volumes -enclosing the region closest to each point in the set. In two -dimensions, the vertices and edges of these polygons form a -tri-coordinate lattice.

    -% -

    % The Kitaev honeycomb lattice model (HLM) is composed of spin-() -particles interacting anisotropically along the edges of a lattice: % -\begin{equation} % = --{(i,j)}J^{{ij}}i^{{ij}}j^{{ij}} + -_{(i,j,k)}_i^{x}j^{y}k^{z} % \end{equation} % where the two spin -term runs over pairs of nearest neighbours and the three spin term runs -over consecutive triplets around a plaquette. The Pauli matrices \(\sigma^\alpha\) in each term are chosen -according to the type, or -coloring', of the bond $i\to j$, $\alpha_{ij}\in\{x,y,z\}$. The bond coloring is chosen such that exactly one bond of each type is connected to each vertex. % This Hamiltonian is exactly solvable by introducing a Majorana representation \(\widetilde{\sigma}_i^{\alpha} = i b^{\alpha}_i c_i\) which the partitions the Hilbert space into a classical $\mathrm{Z}_2$ gauge degree of freedom, \(u_{jk} = ib_j^{\alpha_{jk}}b_k^{\alpha_{jk}}\), on the bonds, and Majorana fermions, $c_j$, living on the vertices. It also doubles the size of the Fock space, necessitating calculating a projector \(P\) from the Majorana Fock space \(\mathcal{\widetilde{M}}\) onto the physical subspace \(\mathcal{M}\)~\cite{pedrocchiPhysicalSolutionsKitaev2011}. We refer to a choice of gauge configuration, $\{u_{jk}\}$, as theflux -sector’. The problem then reduces to solving a free-fermion Hamiltonian -within each flux sector (u) % \begin{equation} % ^u = -{j,k}A{jk}c_jc_k % \end{equation} % where \(A_{jk}=2J^\alpha_{jk}u_{jk}\) for \((j, k)\) nearest-neighbours, \(A_{jk}=2\kappa\sum_l u_{jl}u_{kl}\) for -\((j,k)\) second-nearest-neighbours, -and \(A_{jk}=0\) otherwise. Finally the -Majorana modes can be found with a transformation (Q) % \begin{equation} -% (b^{’}_1, b^{’’}_1, … ;b^{’}_N, b^{’‘}N) = (c_1, c_2, … -;c{2N}) Q % \end{equation} % from which we create the fermionic -operators (a_i = (b^{’}_i + ib^{’’}_i)), bringing (H) to the form % -\begin{equation} % ^u = _m _m (n_m - ) % \end{equation} % with ground -state energy (E_0 = -_m m). The projector has the effect of removing -many body states with either even or odd parity (= i (1 - 2n_i)), an -effect which typically leads to a correction of order (). The gauge -symmetries of \(\{u_{jk}\}\) can be -removed by defining plaquette operators (P_i = {(i,j) P_i} -u{ij}) that wind the plaquettes (faces) of the lattice.

    -

    % The ground state flux sector of the HLM in the isotropic phase -(\(J^x = J^y = J^z\)) at zero field -(\(\kappa=0\)) possesses a gapless -fermionic spectrum. A non-zero field (\(\kappa\neq0\)) opens a gap, and the -resulting fermionic insulator is known to host non-Abelian anyonic -excitations and possess a non-zero Chern number~. This non-abelian phase -has been shown to undergo a finite-temperature phase transition to a -so-called `thermal metal’ phase, which exhibits multifractility~.

    -

    For a lattice with (B) bonds, (V) vertices, (P) plaquettes and Euler -characteristic () (0 for the torus) the Euler equation states that (B = -P + V + ). This corresponds to the (2^{B}) gauge configurations being -composed of (2^{P - 1}) physically distinct vortex states each of which -is composed of (2^{V - 1}) gauge equivalent states that correspond to -flipping three (u_{ij}) around a vertex, along with (2 - ) -non-contractible loop operators. The term (2 - ) is perhaps more easily -understood by relating () to the genus of the surface (g), i.e the -number of holes with (= 2 - 2g) showing that there are two -non-contractible loops for each hole in the surface.

    -

    Care must be taken in the definition of open boundary conditions, -simply removing bonds from the lattice leaves behind unpaired (b^) -operators that need to be paired in some way to arrive at fermionic -modes. In order to fix a pairing we always start from a lattice defined -on the torus and generate a lattice with open boundary conditions by -defining the bond coupling (J^{}_{ij} = 0) for sites joined by bonds -((i,j)) that we want to remove. This creates fermionic zero modes (u_ij) -associated with these cut bonds which we set to 1 when calculating the -projector.

    -

    { Add brief mention of fermions and many body ground state} Closely -following the derivation of~ we can extend to the amorphous case -relatively simply. The main quantity needed is the product of the local -projectors (D_i) [_i^{2N} D_i = _i^{2N} b^x_i b^y_i b^z_i c_i ] for a -lattice with (2N) vertices and (3N) edges. The operators can be ordered -by bond type without utilising any property of the lattice. [_i^{2N} D_i -= _i^{2N} b^x_i _i^{2N} b^y_i _i^{2N} b^z_i _i^{2N} c_i] The product -over (c_i) operators reduces to a determinant of the Q matrix and the -fermion parity. The only problem is to compute the factors (p_x,p_y,p_z -= ) that arise from reordering the b operators such that pairs of -vertices linked by the corresponding bonds are adjacent. [i^{2N} -b^i = p{(i,j)}b^_i b^_j] This is simple the parity of the -permutation from one ordering to the other and can be computed easily -with a cycle decomposition.

    -

    The final form is almost identical to the honeycomb case with the -addition of the lattice structure factors (p_x,p_y,p_z) [P^0 = 1 + -p_x;p_y;p_z (Q^u) ; ; {{i,j}} -iu{ij}]

    -

    ((Q^u)) is the determinant of the matrix that takes ((c_1, c_2… -c_{2N}) Q = (b_1, b_2… b_{2N})). This along with (u_{ij}) depend on the -lattice and the particular vortex sector.

    -

    ( = ^{N} (1 - 2_i)) is the parity of the particular many body state -determined by fermionic occupation numbers (n_i). The Hamiltonian is (H -= _i (n_i - 1/2)) in this basis and this tells use that the ground state -is either an empty system with all (n_i = 0) or a state with a single -fermion in the lowest level.

    -In this section we detail the numerical evidence collected to support -the claim that, for an arbitrary lattice, a gapped ground state flux -sector is found by setting the flux through each plaquette to \(\phi_{\textup{g.s.}} = -(\pm -i)^{n_{\textup{sides}}}\). This was done by generating a large -number (\(\sim\) 25,000) of lattices -and exhaustively checking every possible flux sector to find the -configuration with the lowest energy. We checked both the isotropic -point (\(J^\alpha = 1\)), as well as in -the toric code phase (\(J^x = J^y = 0.25, J^z -= 1\)). -The argument has one complication: for a graph with \(n_p\) plaquettes, there are \(2^{n_p - 1}\) distinct flux sectors to -search over, with an added factor of 4 when the global fluxes \(\Phi_x\) and \(\Phi_y\) are taken into account. Note that -the \(-1\) appears in this counting -because fluxes can only be flipped in pairs. To be able to search over -the entire flux space, one is necessarily restricted to looking at small -system sizes – we were able to check all flux sectors for systems with -\(n_p \leq 16\) in a reasonable amount -of time. However, at such small system size we find that finite size -effects are substantial enough to destroy our results. In order to -overcome these effects we tile the system and use Bloch’s theorem (a -trick that we shall refer to as for reasons that shall become clear) to -efficiently find the energy of a much larger (but periodic) lattice. -Thus we are able to suppress finite size effects, at the expense of -losing long-range disorder in the lattice. -

    Our argument has three parts: First we shall detail the techniques -used to exhaustively search the flux space for a given lattice. Next, we -discuss finite-size effects and explain the way that our methods are -modified by the twist-averaging procedure. Finally, we demonstrate that -as the size of the disordered system is increased, the effect of -twist-averaging becomes negligible – suggesting that our conclusions -still apply in the case of large disordered lattices.

    -{} For a given lattice and flux sector, defined by \(\{ u_{jk}\}\), the fermionic ground state -energy is calculated by taking the sum of the negative eigenvalues of -the matrix \[\begin{align} - M_{jk} = \frac{i}{2} J^{\alpha} u_{jk}. -\end{align}\] The set of bond variables \(u_{jk}\), which we are free to choose, -determine the \(\mathbb Z_2\) gauge -field. However only the fluxes, defined for each plaquette according to -eqn.~\(\ref{eqn:flux_definition}\), -have any effect on the energies. Thus, there is enormous degeneracy in -the \(u_{jk}\) degrees of freedom. -Flipping the bonds along any closed loop on the dual lattice has no -effect on the fluxes, since each plaquette has had an even number of its -constituent bonds flipped - as is shown in the following diagram: -where the flipped bonds are shown in red. In order to explore every -possible flux sector using the \(u_{jk}\) variables, we restrict ourselves -to change only a subset of the bonds in the system. In particular, we -construct a spanning tree on the dual lattice, which passes through -every plaquette in the system, but contains no loops. -

    The tree contains \(n_p - 1\) edges, -shown in red, whose configuration space has a \(1:1\) mapping onto the \(2^{n_p - 1}\) distinct flux sectors. Each -flux sector can be created in precisely one way by flipping edges only -on the tree (provided all other bond variables not on the tree remain -fixed). Thus, all possible flux sectors can be accessed by iterating -over all configurations of edges on this spanning tree.

    -{} In our numerical investigation, the objective was to test as many -example lattices as possible. We aim for the largest lattice size that -could be efficiently solved, requiring a balance between lattice size -and cases tested. Each added plaquette doubles the number of flux -sectors that must be checked. 25,000 lattices containing 16 plaquettes -were used. However, in his numerical investigation of the honeycomb -model, Kitaev demonstrated that finite size effects persist up to much -larger lattice sizes than we were able to access . -In order to circumvent this problem, we treat the 16-plaquette amorphous -lattice as a unit cell in an arbitrarily large periodic system. The -bonds that originally connected across the periodic boundaries now -connect adjacent unit cells. This infinite periodic Hamiltonian can then -be solved using Bloch’s theorem, since the larger system is diagonalised -by a plane wave ansatz. For a given crystal momentum $\bf q\(. We then check if the lowest energy flux sector -aligns with our ansatz (eqn.~\ref{eqn:gnd_flux}) and whether this flux -sector is gapped. \par In the isotropic case (\)J^= 1\(), all 25,000 examples conformed to our guess for -the ground state flux sector. A tiny minority (\)10$) of the -systems were found to be gapless. As we shall see shortly, the -proportion of gapless systems vanishes as we increase the size of the -amorphous lattice. An example of the energies and gaps for one of the -systems tested is shown in fig.~\(\ref{fig:energy_gaps_example}\). For the -anisotropic phase (we used $ J^x, J^y = 0.25, J^z = 1\() the overwhelming majority of cases adhered to -our ansatz, however a small minority (\)0.5 %$) did not. In these -cases, however, the energy difference between our ansatz and the ground -state was at most of order \(10^{-6}\). -Further investigation would need to be undertaken to determine whether -these anomalous systems are a finite size effect due to the small -amorphous system sizes used or a genuine feature of the toric code phase -on such lattices. -

    {} Now that we have collected sufficient evidence to support our -guess for the ground state flux sector, we turn our attention to -checking that this sector is gapped. We no longer need to exhaustively -search over flux space for the ground state, so it is possible to go to -much larger system size. We generate 40 sets of systems with plaquette -numbers ranging from 9 to 1600. For each system size, 1000 distinct -lattices are generated and the energy and gap size are calculated -without phase twisting, since the effect is negligible for such large -system sizes. As can be seen, for very small system size a small -minority of gapless systems appear, however beyond around 20 plaquettes -all systems had a stable fermion gap in the ground state.

    -% Thus, we shall begin with a discussion of how finite size affects the -eigenvalues of the Majorana Hamiltonian, followed by our solution to -this problem. Evidence for the ground state solution was collected by -searching over all possible flux sectors for the lowest energy states. -This is repeated for various values of \(J\) over a large number of randomly -generated lattices. -

    % For a given lattice and flux sector, defined by \(\{ u_{jk}\}\), the fermionic ground state -energy is found by taking the sum of the negative eigenvalues of the -matrix % \begin{align} % M_{jk} = J^{} u_{jk}. % \end{align} % A gauge -transformation involves flipping the value of \(u_{jk}\) for the three bonds connected to -the point at \(j\). Under a gauge -transformation, the matrix \(M\) -transforms according to \(M \rightarrow D_j M -D_j\), where the matrix \(D_j\) -is a diagonal matrix with \(-1\) on the -\(j\)’th entry, and \(+1\) on all others. This represents a -unitary transformation, so the spectrum of \(M\) is invariant under gauge -transformations. As demonstrated in , the spectrum is determined -entirely by the flux through all circuits in the system, which we define -analogously to \(\ref{eqn:flux_definition}\). In this case -we include not only plaquettes, but circuits that encircle several -plaquettes. In periodic boundaries we must also consider

    -% In the language of graph theory, this matrix may be interpreted as -representing a weighted, directed digraph, with weights determined by -the individual entries of \(M\). The -Harary-Sachs theorem states that the characteristic polynomial of such a -matrix may be written in terms of the weights of the cycles of the -graph, defined as the product of the elements of \(M\) around some closed path \(\mathcal C\) on the lattice, % -\begin{align} % w_{} = {} M{jk}. % \end{align} % These weights -are similar to the fluxes defined in the bulk text, with two important -differences. Firstly, the cyclic weights include the factor of \(J^\alpha\) in the product. Secondly, unlike -tthe fluxes, which are defined for individual plaquettes, the weights -are calculated for every closed path on the lattice. The takeaway is -that the characteristic polynomial, and thus all eigenvalues, is -determined only by the values of these weights. Any change to the set of -\(u_{jk}\) that does not affect the -weight of any cycles will have no effect on the energies of the system. -For example a gauge transformation, where \(u_{jk}\) is flipped on the three edges -connected to a chosen site, cannot affect the energies, as every cycle -passing through the chosen site must contain two of the flipped edges. -

    \end{document}

    +

    Chern Markers

    +

    We know that the standard Kitaev model supports both Abelian and +non-Abelian phases, so how can we assess whether this is also the case +for the amorphous Kitaev model?

    +

    We have already discussed the fact that topology and anyonic +statistics are intimately linked and this will help here. The Chern +number is a quantity that measured the topological characteristics of a +material.

    +

    The original definition of the Chern number relies on the model +having translation symmetry. This lead to the development of local +markers, these are operators defined locally that generalise the +notion of the chern number to a local observable over some region +smaller than the entire system.

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    diff --git a/_thesis/2.3_AMK_Results.html b/_thesis/2.3_AMK_Results.html index b8eeccf..0cbf864 100644 --- a/_thesis/2.3_AMK_Results.html +++ b/_thesis/2.3_AMK_Results.html @@ -207,23 +207,18 @@ image:
  • Chapter 2: The Amorphous Kitaev Model
  • Conclusion