Is the $SU(2)$ flux defined in the context of Projective Symmetry Group(PSG) an observable quantity?

The $SU(2)$ flux defined in the context of PSG is as follows:

Consider the mean-field Hamiltonian $H_{MF}=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ description of a 2D lattice spin-model, the definition of $SU(2)$ flux $P_C$ for a loop $C=i\rightarrow j_1\rightarrow j_2\rightarrow ...\rightarrow j_n\rightarrow i$ with the base point $i$ is $P_C=\chi_{ij_1}\chi_{j_1j_2}...\chi_{j_ni}$. On the other hand, the two $SU(2)$ gauge-equivalent mean-field ansatz $\chi_{ij}$ and $\chi_{ij}'=G_i\chi_{ij}G_j^\dagger$ describe the same projected spin-wavefunction. And the $SU(2)$ flux $P_C'$ for the same loop $C$ is given by $P_C'=\chi_{ij_1}'\chi_{j_1j_2}'...\chi_{j_ni}'=G_iP_CG_i^\dagger$, in general $P_C' \neq P_C$, but an observable quantity should be invariant under the $SU(2)$ gauge transformation $\chi_{ij}\rightarrow\chi_{ij}'=G_i\chi_{ij}G_j^\dagger$,. Thus, does this mean that the $SU(2)$ flux $P_C$ is not an observable quantity?

• @ Shenghan Jiang Thanks. And the (scalar) flux through the loop $C$ is defined as the phase of $Tr(P_C)$? – Kai Li Oct 19 '13 at 8:58