2 Background
Anyons are exotic two dimensional particles that are intermediate between bosons and fermions. Abelian anyons pick up an arbitrary phase \(e^{i\phi}\) up interchange. The Kitaev model is also topologically non-trivial, supporting a degenerate ground state manifold of varying size. The interchange of non-Abelian anyons corresponds to arbitrary rotations within the ground state manifold, operations which may not commute and thus form a non-Abelian group.
Anyonic Statistics
NB: I’m thinking about moving this section to the overall intro, but it’s nice to be able to refer to specifics of the Kitaev model also so I’m not sure. It currently repeats a discussion of the ground state degeneracy from the projector section.
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. Therefore, doing one swap can at most multiply it by \(\pm 1\).
However, there are many hidden parts to this argument. First, this argument does not present the whole story. For instance, if you want to know why Fermions have half integer spin, you have to go to field theory.
Second, why does this argument only work in dimensions greater than two? When we say that two swaps do nothing, we in fact say that the world lines of two particles that have been swapped twice can be untangled without crossing. Why can’t they cross? Because if they cross, the particles can interact and the quantum state could change in an arbitrary way. We are implicitly using the locality of physics to argue that, if the worldlines stay well separated, the overall quantum state cannot change.
In two dimensions, we cannot untangle the worldlines of two particles that have swapped places. They are braided together (see fig. 1).
From this fact flows a whole of behaviours. The quantum state can acquire 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 between 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 annihilate them together. Without topology, this should leave the quantum state unchanged. Instead, we move towards 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. In general, these matrices do not commute so these are known as non-Abelian anyons.
From here, the situation becomes even more complex. The Kitaev model has a non-Abelian phase when exposed to a magnetic field. The amorphous Kitaev Model has a non-Abelian phase because of its broken chiral symmetry.
By subdividing the Kitaev model into vortex sectors, we neatly separate between vortices and fermionic excitations. However, if we looked at the full many body picture, we would see that a vortex carries with it a cloud of bound Majorana states.
Consider two processes
We transport one half of a vortex pair around either the x or y loops of the torus before annihilating back to the ground state vortex sector \(\mathcal{T}_{x,y}\).
We flip a line of bond operators corresponding to measuring the flux through either the major or minor axes of the torus \(\mathcal{\Phi}_{x,y}\)
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 at will before bringing them back together. They will annihilate back to the vacuum, where we understand ‘the vacuum’ to refer to one of the ground states. However, this will not necessarily be the same ground state we started in. We can use this to get from the \((\Phi_x, \Phi_y) = (+1, +1)\) ground state and construct the set \((+1, +1), (+1, -1), (-1, +1), (-1, -1)\).
![Figure 2: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make [1].](/assets/thesis/amk_chapter/topological_fluxes.png)
However, in the non-Abelian phase we have to wrangle with monodromy [2,3]. Monodromy is the behaviour of objects as they move around a singularity. This manifests here in that the identity of a vortex and cloud of Majoranas can change as we wind them around the torus in such a way that, rather than annihilating to the vacuum, we annihilate them to create an excited state instead of a ground state. This means that we end up with only three degenerate ground states in the non-Abelian phase \((+1, +1), (+1, -1), (-1, +1)\) [4,5]. Concretely, this is because the projector enforces both flux and fermion parity. When we wind a vortex around both non-contractible loops of the torus, it flips the flux parity. Therefore, we have to introduce a fermionic excitation to make the state physical. Hence, the process does not give a fourth ground state.
Recently, the topology has notably gained interest because of proposals to use this ground state degeneracy to implement both passively fault tolerant and actively stabilised quantum computations [6–8].
What is 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 not scale with system size.
However, there are quantum phases that cannot be characterised by such a local order parameter. These phases are instead said to possess ‘topological order’.
One easily observable property of topological order is 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 is a good example. 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.
Next Section: Disorder and Localisation