diff --git a/_sass/main.scss b/_sass/main.scss index 8f06ccc..441b17f 100644 --- a/_sass/main.scss +++ b/_sass/main.scss @@ -79,6 +79,16 @@ 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/_thesis/1.1_FK_Intro.html b/_thesis/1.1_FK_Intro.html index 6965702..ceeae2b 100644 --- a/_thesis/1.1_FK_Intro.html +++ b/_thesis/1.1_FK_Intro.html @@ -780,8 +780,8 @@ the states in the unperturbed bands remain extended, evidence for a mobility edge.

- +Figure 1: Hello I am the figure caption!
diff --git a/_thesis/2.1.2_AMK_Intro.html b/_thesis/2.1.2_AMK_Intro.html new file mode 100644 index 0000000..826103c --- /dev/null +++ b/_thesis/2.1.2_AMK_Intro.html @@ -0,0 +1,1066 @@ +--- +title: The Amorphous Kitaev Model - Introduction part 2 +excerpt: A short introduction to the weird and wonderful world of exactly solvable quantum models. +layout: none +image: + +--- + + + + + + + + The Amorphous Kitaev Model - Introduction part 2 + + + + + + + + + + + + + + +{% include header.html %} + +
+ +

Properties of the Gauge +Field

+

The bond operators \(u_{ij}\) are +useful because they label a bond sector \(\mathcal{\tilde{L}}_u\) in which we can +easiy solve the Hamiltonian. However the gauge operators move us between +bond sectors. Bond sectors are not gauge invariant!

+

Let’s consider instead the properties of the plaquette operators +\(\hat{\phi}_i\) that live on the faces +of the lattice.

+

We already showed that they are conserved. And as one might hope and +expect, the plaquette operators map cleanly on to the bond operators of +the Majorana representation:

+

\[\begin{aligned} +\tilde{W}_p &= \prod_{\mathrm{i,j}\; \in\; p} \tilde{K}_{ij}\\ + &= \prod_{\mathrm{i,j}\; \in\; p} +\tilde{\sigma}_i^\alpha \tilde{\sigma}_j^\alpha\\ + &= \prod_{\mathrm{i,j}\; \in\; p} (ib^\alpha_i +c_i)(ib^\alpha_j c_j)\\ + &= \prod_{\mathrm{i,j}\; \in\; p} i u_{ij} c_i c_j\\ + &= \prod_{\mathrm{i,j}\; \in\; p} i u_{ij} +\end{aligned}\]

+

Where the last steps holds because each \(c_i\) appears exactly twice and adjacent to +its neighbour in each plaquette operator. Note that this is consistent +with the observation from earlier that each \(W_p\) takes values \(\pm 1\) for even paths and \(\pm i\) for odd paths.

+

Vortices and their movements

+

Let’s imagine we started from the ground state of the model and +flipped the sign of a single bond. In doing so we will flip the sign of +the two plaquettes adjacent to that bond. I’ll call these disturbed +plaquettes vortices. I’ll refer to a particular choice values +for the plaquette operators as a vortex sector.

+

If we chain multiple bond flips we can create a pair of vortices at +arbitrary locations. The chain of bonds that we must flip corresponds to +a path on the dual of the lattice.

+

Something else we can do is create a pair of vortices, move one +around a loop and then anhilate it with its partner. This corresponds to +a closed loop on the dual lattice and applying such a bond flip leaves +the vortex sector unchanged.

+

Notice that the \(D_j\) operators +flip three bonds around a vertex. This is the smallest closed loop +around which one can move a vortex pair and anhilate it with itself.

+

Such operations compose in the sense that we can build any larger +loop by applying a series of \(D_j\) +operations. Indeed the symetrisation procedure \(\prod_i \left( \frac{1 + D_i}{2}\right)\) +that maps from the bond sector to a physical state is applying +constructing a superposition over every such loop that leaves the vortex +sector unchanged.

+

The only loops that we cannot build out of \(D_j\)s are non-contractible loops, such as +those that span the major or minor circumference of the torus.

+

The plaquette operators are the gauge invariant quantity that +determines the physics of the model

+

Composition of \(u_{jk}\) loops

+
+
+ + +
+
+

Second it is now easy to show that the loops and plaquettes satisfy +nice composition rules, so long as we stick to loops that wind in a +particular direction.

+

Consider the product of two non-overlapping loops \(W_a\) and \(W_b\) that share an edge \(u_{12}\). Since the two loops both wind +clockwise and do not overlap, one will contain a term \(i u_{12}\) and the other \(i u_{21}\). Since the \(u_{ij}\) commute with one another, they +square to \(1\) and \(u_{ij} = -u_{ji}\) we see have \(i u_{12} i u_{21} = 1\) and we can repeat +this for any number of shared edges. Hence, we get a version of Stokes’ +theorem: the product of \(i u_{jk}\) +around any closed loop \(\partial A\) +is equal to the product of plaquette operators \(\Phi\) that span the area \(A\) enclosed by that loop: \[\prod_{u_{jk} \in \partial A} i \; u_{jk} = +\prod_{\phi_i \in A} \phi_i\]

+
+
+ + +
+
+

Wilson loops can always be decomposed into products of +plaquettes operators unless they are non-contractable

+

Gauge Degeneracy and +the Euler Equation

+

We can check this analysis with a counting argument. For a lattice +with \(B\) bonds, \(P\) plaquettes and \(V\) vertices we can count how many bond +sectors, vortices sectors and gauge symmetries there are and check them +against Euler’s polyhedra equation.

+

Euler’s equation states for a closed surface of genus \(g\), i.e that has \(g\) holes so \(0\) for the sphere, \(1\) for the torus and \(g\) for \(g\) tori stuck together \[B = P + V + 2 - 2g\]

+
+
+ + +
+
+

For the case of the torus where \(g = +1\) we can rearrange this to read: \[B += (P-1) + (V-1) + 2\]

+

Each \(u_{ij}\) takes two values and +there is one associated with each bond so there are exactly \(2^B\) distinct configurations of the bond +sector. Let’s see if we can factor those configurations out into the +cartesian product of vortex sectors, gauge symmetries and +non-contractible loop operators.

+

Vortex sectors: each plaquette operator \(\phi_i\) takes two values (\(\pm 1\) or \(\pm +i\)) and there are \(P\) of them +so naively one would think there are \(2^P\). However vortices can only be created +on pairs so there are really \(\tfrac{2^P}{2} += 2^{P-1}\) vortex sectors.

+

Gauge symmetries: As discussed earlier these correspond to the all +possible compositions of the \(D_j\) +operators. Again there are only \(2^{V-1}\) of these because, as we will see +in the next section, \(\prod_{j} D_j = +\mathbb{1}\) in the physical space, and we enforce this by +chooising the correct product of single particle fermion states. You can +get an intuitive picture for why \(\prod_{j} +D_j = \mathbb{1}\) by imagining larger and larger patches of +\(D_j\) operators on the torus. These +patches correspond to transporting a vortex pair around the edge of the +patch. At some point the patch wraps around and starts to cover the +entire torus, as this happens the bounday of the patch disappears and +hence it which corresponds to the identity operation. See Fig ?? +(animated in the HTML version).

+

Finally the torus has two non-contractible loop operators asscociated +with its major and minor diameters.

+

Putting this all together we see that there are \(2^B\) bond sectors a space which +can be decomposed into the cartesian product of \(2^{P-1}\) vortex sectors, +\(2^{V-1}\) gauge +symmetries and \(2^2 = +4\) topological sectors associated with the +non-contractible loop operators. This last factor forms the basis of +proposals to construct topologically protected qubits since the 4 +sectors cold only be mixed by a highly non-local perturbation, ref +?????.

+

+ + +

+

Counting edges, +plaquettes and vertices

+

It will be useful to know how the trivalent structre of the lattice +constraints the number of bonds \(B\), +plaquettes \(P\) and vertices \(V\) it has.

+

We can immediately see that the lattice is built from vertices that +each share 3 edges with their neighbours. This means each vertex comes +with \(\tfrac{3}{2}\) bonds i.e \(3V = 2B\). This is consistent with the fact +that in the Majorana representation on the torus each vertex brings +three \(b^\alpha\) operators which then +pair along bonds to give \(3/2\) bonds +per vertex.

+

If we define an integer \(N\) such +that \(V = 2N\) and \(B = 3N\) and substitite this into the +polyhedra equation for the torus we see that \(P = N\). So if is a trivalent lattice on +the torus has \(N\) plaquettes, it has +\(2N\) vertices and \(3N\) bonds.

+

We can also consider the sum of the number of bonds in each plaquette +\(S_p\), since each bond is a member of +exactly two plaquettes \[S_p = 2B = +6N\]

+

The mean size of a plaquette in a trivalent lattice on the torus is +exactly 6. Since the sum is even, this also tells us that all odd +plaquettes must come in pairs.

+
+
+ + +
+
+

The Projector

+

It will turn out that the projection from the extended space to the +physical space is not actually that important for the results that I +will present. However it it useful to go through the theory of it to +explain why this is.

+

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

+

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

+

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

+

\[ \mathcal{P} = \frac{1}{2^{2N}} +\sum_{\{i\}} \prod_{i\in\{i\}} D_i\]

+

Since the gauge operators \(D_j\) +commute and square to one, we can define the complement operator \(C = \prod_{i=1}^{2N} D_i\) and see that it +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 +operator deals with the other half

+

\[ \mathcal{P} = \left( +\frac{1}{2^{2N-1}} \sum_{\Lambda} \prod_{i\in\{i\}} D_i\right) +\left(\frac{1 + \prod_i^{2N} D_i}{2}\right) = \mathcal{S} \cdot +\mathcal{P}_0\]

+

To compute \(\mathcal{P}_0\) the +main quantity needed is the product of the local projectors \(D_i\) \[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i b^y_i b^z_i +c_i \] for a toroidal trivalent lattice with \(N\) plaquettes \(2N\) vertices and \(3N\) edges.

+

First we reorder the operators by bond type, this doesn’t require any +information about the underlying lattice.

+

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

+

The product over \(c_i\) operators +reduces to a determinant of the Q matrix and the fermion parity, +see1 . 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 +corresponding bonds are adjacent.

+

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

+

However they are simply the parity of the permutation from one +ordering to the other and can be computed in linear time with a cycle +decompositionapp:cycle_decomp?.

+

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

+

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 \(c_i\) 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. \(\hat{\pi} = \prod{i}^{N} (1 - +2\hat{n}_i)\) is the parity of the particular many body state +determined by fermionic occupation numbers \(n_i\).

+

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

+

Let’s think about where are with the model now. We can map the spin +Hamiltonian to a Majorana Hamiltonian in an extended Hilbert space. +Along with that mapping comes a gauge field \(u_{jk}\) defining bond +sectors. The gauge symmetries of \(u_{jk}\) are generated by the set of \(D_j\) operators. The gauge invariant and +therefore physically relevant variables are the plaquette operators +\(\phi_i\) which define as a +vortex sector. In order to practically solve the +Majorana Hamiltonian we must remove hats from the gauge field by +restricting ourselves to a particular bond sector. From there the +Majorana Hamiltonian becomes non-interacting and we can solve it like +any quadratic theory. This lets us construct the single particle +eigenstates from which we can also construct many body states. However +the many body states constructed this way are not in the physical +subspace!

+

However for the many body states within a particular bond sector, +\(\mathcal{P}_0 = 0,1\) tells us which +of those have some overlap with the physical sector.

+

We see that finding a state that has overlap with a physical state +only ever requires the addition or removal of one fermion. There are +cases where this can make a difference but for most observables such as +ground state energy this correction scales away as the number of +fermions in the system grows.

+

If we wanted to construct a full many body wavefunction in the spin +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

+

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

+

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 +evidence that this guess is in fact correct.

+

Let’s consider the partition function of the Majorana hamiltonian: +\[ \mathcal{Z} = \mathrm{Tr}\left( e^{-\beta +H}\right) = \sum_i \exp{-\beta \epsilon_i}\] At low temperatures +\(\mathcal{Z} \approx \beta +\epsilon_0\) where \(\epsilon_0\) is the lowest energy fermionic +state.

+

How does the \(\mathcal{Z}\) depend +on the Majorana hamiltonian? Expanding the exponential out gives: \[ \mathcal{Z} = \sum_n \frac{(-\beta)^n}{n!} +\mathrm{Tr(H^k)} \]

+

Now there’s an interesting observation to make here. The Hamiltonian +is essentially a scaled adjacency matrix. An adjacency matrix being a +matrix \(g_{ij}\) such that \(g_{ij} = 1\) if vertices \(i\) and \(j\) and joined by an edge and 0 +otherwise.

+

Powers of adjacency matrices have the property that the entry \((g^n)_{ij}\) corresponds to the number of +paths of length n on the graph that begin at site \(i\) and end at site \(j\). These include somewhat degenerate +paths that go back on themselves etc.

+

The trace of an adjacency matrix \[\mathrm{Tr}(g^n) = \sum_i (g^n)_{ii}\] +therefore counts the number number of loops of size \(n\) that can be drawn on the graph.

+

Applying the same treatment to our Majorana Hamiltonian, we can +interpret \(u_ij\) to equal 0 if the +two sites are not joined by a bond and we put ourselves in the isotropic +phase where \(J^\alpha = 1\) \[ \tilde{H}_{ij} = \tfrac{1}{2} i +u_{ij}\]

+

We then see that the trace of the nth power of H is a sums over +Wilson loops of size \(n\) with an +additional factor of \(2^{-n}\). We +showed earlier that the Wilson loop operators can always be written as +products of the plaquette operators that they enclose.

+

Lumping all the prefactors together, we can write: \[ \mathcal{Z} = c_A \hat{A} + c_B \hat{B} + \sum_i +c_i \hat{\phi}_i + \sum_{ij} c_{ij} \hat{\phi}_i \hat{\phi}_j + +\sum_{ijk} c_{ijk} \hat{\phi}_i \hat{\phi}_j \hat{\phi}_k + +...\]

+

Where the \(c\) factors would be +something like \[c_{ijk...} = \sum_n +\tfrac{(-\beta)^n}{n!} \tfrac{1}{2^n} K_{ijk...}\] which is a sum +over all loop lengths \(n\) and for +each we have a combinatoral factor \(K_{ijk...}\) that counts how many ways +there are to draw a loop of length \(n\) that only encloses plaquettes \(ijk...\).

+

We also have the pesky non-contractible loop operators \(\hat{A}\) and \(\hat{B}\). Again the prefactors for these +are very complicated but we can intuitively see that for larger and +larger loops lengths there will be a combinatorial explosion of possible +ways that they appear in these sums. These are suppressed exponentially +with system size but at practical lattice sizes they cause significant +finite size effects. The main evidence of this is that the 4 loop +sectors spanned by the \(\hat{A}\) and +\(\hat{B}\) operators are degenerate in +the infinite system size limit, while that degeneracy is lifted in +finite sized systems.

+

We don’t have much hope of actually evaluating this for an amorphous +lattice. However it lead us to guess that the ground state vortex sector +might be a simple function of the side length of each plaquette.

+

The ground state of the Amorphous Kitaev Model is found by setting +the flux through each plaquette \(\phi\) to be equal to \(\phi^{\mathrm{g.s.}}(n_{\mathrm{sides}})\)

+

\[\begin{aligned} + \phi^{\mathrm{g.s.}}(n_{\mathrm{sides}}) = -(\pm +i)^{n_{\mathrm{sides}}}, +\end{aligned}\] where \(n_{\mathrm{sides}}\) 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 +lattices2 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.

+

Finite size effects

+

This guess only works for larger lattices because of the finite size +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 +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 +non-contractible loop operators are removed with only a linear penalty +in computation time compared to the exponential penalty paid by simply +simply diagonalising larger lattices.

+

Using this technique we verified that \(\phi_0\) 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

+

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 model3–10.

+

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 +other by inverting the imaginary \(\phi\) fluxes4.

+

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 +not scale with system size.

+

However there are quantum phases that cannot be characterised by such +a local order parameter. These phases are intead said to posess +‘topological order’.

+

One property of topological order that is particularly easy to +observe that the ground state degeneracy depends on the topology of the +manifold that we put the system on to. This is referred to as +topological degeneracy to distinguish it from standard symmetry +breaking.

+

The Kitaev model will be a good example of this, we have already +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

+

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 +swapped. We call these Fermions and Bosons.

+

This argument is predicated on the idea that performing two swaps is +equivalent to doing nothing. Doing nothing should not change the quantum +state at all, so doing one swap can at most multiply it by \(\pm 1\).

+

However there are many hidden parts to this argument. Firstly, this +argument just isn’t the whole story, if you want to know why Fermions +have half integer spin, for instance, you have to go to field +theory.

+

There is also a second niggle, why does this argument only work in +dimensions greater than two? What we’re really saying when we say that +two swaps do nothing is that the world lines of two particles that have +been swapped twice can be untangled without crossing. Why can’t they +cross? Well because if they cross then the particles can interact and +the quantum state could change in an arbitrary way. We’re implcitly +using the locality of physics here to argue that if the worldlines stay +well separated then the overall quantum state cannot too much.

+

In two dimensions we cannot untangle the worldlines of two particles +that have swapped places, they are braided together. See fig. 5 for a diagram.

+
+
+Figure 5: + +
+
+

From this fact flows a whole new world of behaviours, now the quantum +state can aquire a phase factor \(e^{i\phi}\) upon exchange of two identical +particles, which we now call Anyons.

+

The Kitaev Model is a good demonstration of the connection beween +Anyons and topological degeneracy. In the Kitaev model we can create a +pair of vortices, move one around a non-contractable loop \(\mathcal{T}_{x/y}\) and then anhilate them +together. Without topology this should leave the quantum state +unchanged. Instead it moves us to another ground state in a +topologically degenerate ground state subspace. Practically speaking it +flips a dual line of bonds \(u_{jk}\) +going around the loop which we cannot undo with any gauge transformation +made from \(D_j\) operators.

+

If the ground state subspace is multidimensional, quasiparticle +exchange can move us around in the space with an action corresponding to +a matrix. These matrices do not in general commmute and so these are +known as non-Abelian anyons.

+

From here things get even more complex, the Kitaev model has a +non-Abelian phase when exposed to a magnetic field, and the amorphous +Kitaev Model has a non-Abelian phase because of its broken chiral +symmetry.

+

The way that we have subdivided the Kitaev model into vortex sectors, +we have a neat separation beween vortices and fermionic excitations. +However if we looked at the full many body picture we would see that a +vortex caries with it a cloud of bound majorana states.

+
+
+ + +
+
+

Consider two processes

+
    +

  1. We transport one half of a vortex pair around either the x or y +loops of the torus before anhilating back to the ground state vortex +sector \(\mathcal{T}_{x,y}\).

    2. +

  2. We flip a line of bond operators coresponding to measuring the +flux through either the major or minor axes of the torus \(\mathcal{\Phi}_{x,y}\)

    3. +
+
+
+ + +
+
+

The plaquette operators \(\phi_i\) +are associated with fluxes. Wilson loops that wind the torus are +associated with the fluxes through its two diameters \(\mathcal{\Phi}_{x,y}\).

+

In the Abelian phase we can move a vortex along any path we like and +then when we bring them back together they will anhilate back to the +vacuum, where we understand ‘the vacuum’ to refer to one of the ground +states, though not necesarily the same one we started in. We can use +this to get from the \((\Phi_x, \Phi_y) = (+1, ++1)\) ground state and construct the set \((+1, +1), (+1, -1), (-1, +1), (-1, +-1)\).

+
+
+ + +
+
+

However in the non-Abelian phase we have to wrangle with +monodromy12,13. 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 +vacuum the anhilate to create an excited state instead of a ground +state. This means we end up with only three degenerate ground states in +the non-Abelian phase \((+1, +1), (+1, -1), +(-1, +1)\)14. 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 +fermionic excitation to make the state physical. Hence the process does +not give a fourth ground state.

+

+

One reason the topology has gained interest recently is there have +proposals to use this ground state degeneracy to implement both +passively fault tolerant and actively stabilised quantum computations +[15;16; +hastingsDynamicallyGeneratedLogical2021].

+
+
+
1.
Pedrocchi, F. L., Chesi, S. & Loss, D. Physical solutions of +the Kitaev honeycomb model. Phys. Rev. B +84, 165414 (2011).
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Lieb, E. H. Flux +Phase of the Half-Filled Band. +Physical Review Letters 73, 2158–2161 +(1994).
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Yao, +H. & Kivelson, S. A. An exact chiral +spin liquid with non-Abelian anyons. Phys. Rev. +Lett. 99, 247203 (2007).
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Chua, V. & Fiete, G. A. Exactly solvable +topological chiral spin liquid with random exchange. Phys. Rev. +B 84, 195129 (2011).
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+
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Fiete, G. A. et al. Topological +insulators and quantum spin liquids. Physica E: Low-dimensional +Systems and Nanostructures 44, 845–859 +(2012).
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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).
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+ + diff --git a/_thesis/2.1_AMK_Intro.html b/_thesis/2.1_AMK_Intro.html index ffdff2b..8427b02 100644 --- a/_thesis/2.1_AMK_Intro.html +++ b/_thesis/2.1_AMK_Intro.html @@ -207,10 +207,10 @@ image: @@ -312,6 +278,7 @@ 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

@@ -348,10 +315,10 @@ amorphous case, the key property that will remain is that each vertex interacts with exactly three others via an x, y and z edge. However the lattice will no longer be bipartite, breaking chiral symmetry among other things.

-

Kitaev-Heisenberg Model

-

In real materials there will generally be an addtional small -Heisenberg term \[H_{KH} = - \sum_{\langle -j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} + +

Kitaev-Heisenberg Model In real materials there will generally be an +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

@@ -447,7 +414,7 @@ class="math inline">\(\alpha\) is a function of \(i,j\) that picks the correct bond type.

-Figure 1:
@@ -486,7 +453,7 @@ i\).

Figure 2: The eigenvalues of a loop or plaquette operators depend on how many bonds in its enclosing path.