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.

This post imported from StackExchange Physics at 2014-03-09 08:40 (UCT), posted by SE-user K-boy