The gap magnitude in the anisotropic Kitaev model with ordered-flux structure

Quantum spin liquids in the Kitaev quantum spin model have attracted current interest. This model is exactly solvable and possesses two types of elementary excitations: itinerant Majorana fermions and localized fluxes. In the ground state with a flux free configuration, the Majorana fermion system is gapless while spin correlations are extremely short ranged. In our previous paper, we reported that the Majorana excitation becomes gapful when the system has a certain ordered-flux structure. This may mean that the motion of Majorana fermions is controllable. Thus, it is instructive to examine how the anisotropy in the Kitaev couplings as well as the flux-configurations affect the Majorana excitations since the Kitaev couplings may be changed by the optical irradiation, for example.
To this end, we treat the anisotropic S=1/2 Kitaev model with some flux-ordered configurations. It is known that the Kitaev system is reduced to the toric code model with gapped excitations when one of the Kitaev couplings is large enough. In this study, we diagonalize the Hamiltonian with fixed flux-configurations and discuss how the toric code state is stable in the parameter space. In particular, we clarify how the gapped state realized in an ordered-flux configuration competes with the toric code.