# What is the first excited state of the honeycomb Kitaev model in its gapped phase?

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.

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