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For example, the ground state of the antiferromagnetic(AFM) Heisenberg model $H=J\sum_{<ij>}\mathbf{S}_i \cdot \mathbf{S}_j(J>0)$ on a 2D square lattice is a Neel state, which is a classical order described by conventional order parameters $\left \langle \mathbf{S}_i \right \rangle$.

On the other hand, if we use the Schwinger-fermion mean-field theory to study $H$, we will get a mean-field Hamiltonian $H_{MF}=\sum (f_i^\dagger \chi_{ij} f_j+f_i^\dagger \eta_{ij} f_j^\dagger+H.c.)$ which is studied in Wen's papers of PSG, and now we can study the PSG of the mean-field ansatz $(\chi_{ij},\eta_{ij})$ and associated quantum order.

So according to the above example, can we say that the AFM SDW phase(classical order) possesses the quantum order(PSG)? Thank you very much.

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In my books the classical orders emerge from the quantum mechanical orders. In other words the basic level is quantum mechanical and the classical is a limiting case macroscopically. Coexist does not describe the relationship accurately. –  anna v Sep 5 '13 at 9:45
@ anna v, can you give me a link to your books? Thank you. –  Kai Li Sep 5 '13 at 13:14
I am sorry, it is just an english expession "in my books". I can give you a link to Motl's blog where he shows how from photons one gets the classical electromagnetic field: motls.blogspot.gr/2011/11/… –  anna v Sep 5 '13 at 15:00
I just found a nice work by Prof.Daghofer, which shows the coexistence of both topological order and classical order. –  Kai Li Jan 18 at 11:26

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