Here our arguments are restricted to the realm of the Projective Symmetry Group(PSG) proposed by Prof. Wen,

Quantum Orders and Symmetric Spin Liquids. Xiao-Gang Wen. *Phys. Rev. B* **65** no. 16, 165113 (2002). arXiv:cond-mat/0107071.

and the following notations are the same as those in my previous question, Two puzzles on the Projective Symmetry Group(PSG)?.

When we say the projected physical spin state $P\Psi$ has some 'symmetry', e.g., translation symmetry, there will be **two** understandings:

(1) After a **translation of the mean-field Hamiltonian $H(\psi_i)$**, say $DH(\psi_i)D^{-1}$, the physical spin state is unchanged, say $P\Psi'\propto P\Psi$, where $\Psi'$ is the ground state of the translated Hamiltonian $DH(\psi_i)D^{-1}$.

(2) $D(P\Psi)\propto P\Psi$.

I would like to know: **are the above understandings equivalent to each other?** Thanks in advance.

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