As we saw in the FQH systems, excitations with fractional charge and statistics was really the hall-mark of topologically degenerate states. Since the basic degree of freedom in the toric code are spin, we expect all excitations to be neutral. But there is a possibility that we get fractionalized statistics. The neat thing about the toric code Hamiltonian is that it allows us to not only compute the ground state for the toric code but also all the excited states. Again, this is not too surprising since all the terms in the Hamiltonian commute, so all eigenstates are simultaneous eigenstates of the vertex and plaquette terms. If we focus on the vertex terms first (let’s say by assuming that \(B_p\ll A_v\)), we can get excitations of the vertex Hamiltonian by breaking loops. We can think of the end points of the loops as excitations, since the plaquette terms proportional to \(B_p\) make the plaquette terms fluctuate. These particles (that you see in the figure below) because of analogy with \(Z_2\) gauge theory, are called the electric defects, which we label ‘e’. As shown below, analogous defects in \(\sigma_x=-1\)-loops on the dual lattice are referred to as magnetic defects, which we will label ‘m’.

While the intuitive picture for the excitations as ends of broken loops is nice, to describe these exctiation in the more general case, where \(A_v\) and \(B_p\) are comparable, it is convenient to define the so-called Wilson path operators

\[W_e=\prod_{\mathcal{l}_e} \sigma_z,\quad\, W_m=\prod_{\mathcal{l}_m} \sigma_x.\]

By viewing the system in the \(\sigma_z\)-basis in the limit \(B_p\rightarrow 0\), we see that the operator \(W_e\) counts the parity of \(\sigma_z=-1\) spins that lie on the loop \(\mathcal{l}_e\). Therefore, in this limit \(W_e\) measures the parity of ‘e’ excitations inside the loop \(\mathcal{l}_e\). The operator \(W_e\) is a product of the vertex terms inside the loop \(\mathcal{l}_e\) and hence commutes with \(H\) for any strength of the plaquette terms proportional to \(B_p\).

Therefore \(W_e\) and \(W_m\) are conserved ‘flux’ operators that measure the parity of the number of electric and magnetic defects inside the loops \(\mathcal{l}_{e,m}\) respectively.

Thus, the values \(W_{e,m}=-1\) can also be used to define what it means to have a localized ‘e’ or ‘m’ excitation respectively. These defects describe the localized excitations of the toric code. In fact in this model, this excitation on the ground states are localized to exactly one lattice site and may be viewed as point-like particles in a vacuum.

Just like in the quantum Hall effect, we can use the Wilson loops \(W_{e,m}\) to characterize the degenerate ground states of the toric code on a torus. The value of the Wilson loop \(W_e\) counts the parity of intersections of \(\sigma_z=-1\) loops (red lines) crossing the Wilson loop. Therefore, the value of the Wilson loop \(W_e\) along one of the cycles of the torus counts the parity of the \(\sigma_z=-1\) loops crossing it. Since we can draw a pair of commuting Wilson loop \(W_e\), one through each cycle of the torus, the degeneracy of the torus from \(W_e=\pm 1\) is 4. This is exactly what we got from the loop picture.