As we know, there are both gapless and gapped phases of the Kitaev model, and let's fix the couplings $J_x,J_y,J_z$ such that the model being in the gapped phase. My question is, what is the first excited state in the gapped phase, spinon excitations or $Z_2$ vortex excitations?
Consider the ground state energy $E(F)$ of the quadratic fermionic Hamiltonian as a function of the flux conﬁguration $F$, it is known that the zero-flux conﬁguration $F_0$ minimizes the ground state energy $E(F_0)$. Now let $E(F_1)$ be the second minimal ground state energy corresponding to some flux conﬁguration $F_1$, and define $\Delta_1=E(F_1)-E(F_0)$. And let $\Delta_0$ represents the energy gap of the quadratic fermionic Hamiltonian in the zero-flux conﬁguration $F_0$. [In other words, $\Delta_1$ represents the energy gap of $Z_2$ vortex excitations, and $\Delta_0$ is the energy gap of a fermionic spinon excitation.]
Now there are some possibilities: If $\Delta_0<\Delta_1$, then the first excited state of the whole system would correspond to the spinon excitations; if $\Delta_1<\Delta_0$, then the first excited state would correspond to the $Z_2$ vortex excitations. The energy gap $\Delta$ of the whole system should take $\Delta=min(\Delta_0,\Delta_1)$. But the original paper seems not mentioning this point. Does anybody know some related articles? Thank you very much.