Kitaev's honeycomb model, toric code stabilizers and BCS state
by
DrJiri Vala(National University of Ireland at Maynooth)
→
Europe/Stockholm
132:028
132:028
Description
We present a solution of the Kitaev spin model on the honeycomb lattice
and of related topologically ordered spin models. We employ a
Jordan-Wigner type fermionization and find that the Hamiltonian takes a
BCS type form, allowing the system to be solved by Bogoliubov
transformation. Our fermionization does not employ non-physical
auxiliary degrees of freedom and the eigenstates we obtain are
completely explicit in terms of the spin variables. The ground-state is
obtained as a BCS condensate of fermion pairs over a vacuum state which
corresponds to the toric code state with the same vorticity. We show in
detail how to calculate all eigenstates and eigenvalues of the model on
the torus. In particular, we find that the topological degeneracy on the
torus descends directly from that of the toric code, which now supplies
four vacua for the fermions, one for each choice of periodic vs.
anti-periodic boundary conditions. The reduction of the degeneracy in
the non-Abelian phase of the model is seen to be due to the vanishing of
one of the corresponding candidate BCS ground-states in that phase. This
occurs in particular in the fully periodic vortex-free sector. The true
ground-state in this sector is exhibited and shown to be gapped away
from the three partially anti-periodic ground-states whenever the
non-Abelian phase is gapped.