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

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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?

Thanks in advance.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user K-boy
asked Oct 10, 2013 in Theoretical Physics by Kai Li (975 points) [ no revision ]
retagged Mar 9, 2014
In the field theory language, for nonabelian gauge theory, the observable (or Wilson loop) is Tr(...)

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Shenghan Jiang
@ Shenghan Jiang Thanks. And the (scalar) flux through the loop $C$ is defined as the phase of $Tr(P_C)$?

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user K-boy
Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot.

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user Qmechanic

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