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Based on the previous question and the comment in it, imagine two different mean-field Hamiltonians $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ and $H'=\sum(\psi_i^\dagger\chi_{ij}'\psi_j+H.c.)$, we say that $H$ and $H'$ are gauge equivalent if they have the same eigenvalues and the same projected eigenspaces. And the Wilson loop $W(C)$ can be defined as the trace of matrix-product $P(C)$(see the notations here ). Now my questions are:

(1)"$H$ and $H'$ are gauge equivalent" if and only if "$W(C)=W'(C)$ for all loops $C$ on the 2D lattice ". Is this true? How to prove or disprove it?

(2)If the system is on a 2D torus, is $W(L)$ always a positive real number ? Which means that the 'total flux'(the phase of $W(L)$) through the torus is quantized as $2\pi\times integer$, where $L$ is the boundary of the 2D lattice.

(3)If the Hamiltonian contains extra terms, say $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+\psi_i^T\eta_{ij}\psi_j+H.c.+\psi_i^\dagger h_i\psi_i)$, is the Wilson loop still defined as $W(C)=tr(P(C))$?

Thanks a lot.

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