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As we know, there are both gapless and gapped ground states 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, does the first excited state reside in the zero-flux sector?

Consider the ground state energy $E(F)$ for a given 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 $\Delta_0$ represent the energy gap of the quadratic fermionic Hamiltonian in the zero-flux configuration $F_0$.

Now there are some possibilities: If $\Delta_0<\Delta_1$, then the first excited state of the whole system would still correspond to $F_0$; if $\Delta_1<\Delta_0$, then the first excited state would run into $F_1$. 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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