Recently I'm studying PSG and I felt very puzzled about two statements appeared in Wen's paper. To present the questions clearly, imagine that we use the Shwinger-fermion $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$ mean-field method to study the 2D spin-1/2 system, and get a mean-field Hamiltonian $H(\psi_i)=\sum_{ij}(\psi_i^\dagger u_{ij}\psi_j+\psi_i^T \eta_{ij}\psi_j+H.c.)+\sum_i\psi_i^\dagger h_i\psi_i$, where $\psi_i=(f_{i\uparrow},f_{i\downarrow}^\dagger)^T$, $u_{ij}$ and $\eta_{ij}$ are $2\times2$ complex matrices, and $h_i$ are $2\times2$ Hermitian matrices. And the projection to the spin subspace is implemented by projective operator $P=\prod _i(2\hat{n}_i-\hat{n}_i^2)$(Note here $P\neq \prod _i(1-\hat{n}_{i\uparrow}\hat{n}_{i\downarrow})$). My questions are:

(1)How to arrive at Eq.(15) ? Eq.(15) means that, if $\Psi$ and $\widetilde{\Psi}$ are the mean-field ground states of $H(\psi_i)$ and $H(\widetilde{\psi_i})$, respectively, then $P\widetilde{\Psi}\propto P\Psi$, where $\widetilde{\psi_i}=G_i\psi_i,G_i\in SU(2)$. How to prove this statement?

(2)The statement of translation symmetry above Eq.(16), which can be formulated as follows: Let $D:\psi_i\rightarrow \psi_{i+a}$ be the unitary translation operator($a$ is the lattice vector). If there exists a $SU(2)$ transformation $\psi_i\rightarrow\widetilde{\psi_i}=G_i\psi_i,G_i\in SU(2)$ **such that** $DH(\psi_i)D^{-1}=H(\widetilde{\psi_i})$, then the projected spin state $P\Psi$ has translation symmetry $D(P\Psi)\propto P\Psi $, where $\Psi$ is the mean-field ground state of $H(\psi_i)$. How to prove this statement?

I have been struggling with the above two puzzles for several days and still can't understand them. I will be very appreciated for your answer, thank you very much.

This post imported from StackExchange Physics at 2014-03-09 08:42 (UCT), posted by SE-user K-boy