RVB gauge theory and the topological degeneracy in the honeycomb Kitaev model (Q2911011)

In the search for unconventional, anyonic excitations of two-dimensional quantum spin models (with as yet more or less speculative applications to high-temperature superconductivity, fractional quantum Hall effect and quantum computers), Kitaev introduced a model on the honeycomb lattice with exactly solvable spectrum, reducible to a local SU(2)-gauge theory. The model may be re-expressed in terms of Majorana fermions, including gauge-invariant fermions, and gauge operators, quadratic in the fermions, which are only invariant under the center \(\mathbb Z_2\) of \(SU(2)\). The remaining \(\mathbb Z_2\)-gauge symmetry may be essentially quotiented out by the choice of a Jordan-Wigner fermionization along a covering, non-intersecting path. The Hamiltonian (38) is then a quadratic form in Majorana fermions \(\eta\) with coefficients \(\pm 1\) along normal bounds. The ground state is known independently to correspond to configurations which are flux-free on every plaquette; this leaves open the problem of fixing the value of the Wilson loop along two generators of the covering group of the two-torus (also called large gauge transformations), and leads to a four-fold degeneracy of the ground state. On the other hand, through the Jordan-Wigner fermionization, the constraint on the flux is shown to leave out four sectors, in each of which the Hamiltonian is explicitly diagonalizable, giving a new, algebraic proof of the four-fold degeneracy of the ground state. The article proceeds further to prove four-fold degeneracy of \textit{any} eigenstate by exhibiting in the Jordan-Wigner realization four explicit operators changing the value of the Wilson loop associated to the large gauge transformations without changing the values of the fluxes.