Here I have some confusing points about the definition of flux in the projective construction. For example, consider the same mean-field Hamiltonian in my previous question, and assume the $2\times 2$ complex matrix $\chi_{ij}$ has the form $\begin{pmatrix} t_{ij}& \Delta_{ij}\\ \Delta_{ij}^* & -t_{ij}^* \end{pmatrix}$. Consider a loop with $n$ links on the 2D lattice, the flux through this loop can be defined as the phase of $tr(\chi_1 \cdots\chi_n)$, where $\chi_i=\begin{pmatrix} t_i& \Delta_i\\ \Delta_i^* & -t_i^* \end{pmatrix},i=1,2,...,n$ representing the $i$ th link. And due to the identity $\chi_i^*=-\sigma_y\chi_i\sigma_y$, it's easy to show that $[tr(\chi_1 \cdots\chi_n)]^*=(-1)^ntr(\chi_1 \cdots\chi_n)$, which means that for an even loop, the flux is always $0$ or $\pi$; while for an odd loop, the flux is always $\pm\frac{\pi}{2}$. My questions are as follows:

(1)When $\chi_{ij}=\begin{pmatrix} t_{ij}& 0\\ 0 & -t_{ij}^* \end{pmatrix}$, the mean-field Hamiltonian can be rewritten as $H_{MF}=\sum(t_{ij}f_{i\sigma}^\dagger f_{j\sigma}+H.c.)$, if we define the flux through a loop $i\rightarrow j\rightarrow k\rightarrow \cdots\rightarrow l\rightarrow i$ as the phase of $t_{ij}t_{jk}\cdots t_{li}$, then the flux may take any real value in addition to the above only allowed values $0,\pi,\pm\frac{\pi}{2}$. So which definition of flux is correct?

(2)If $tr(\chi_1 \cdots\chi_n)=0$, how we define the flux(now the phase is highly uncertain)?

Thank you very much.


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