# A simple question on the projected wave function?

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For example, consider a spin-1/2 AFM Heisenberg Hamiltonian $H=\sum_{<ij>}\mathbf{S}_i\cdot\mathbf{S}_j$, and we perform a Schwinger-fermion($\mathbf{S}_i=\frac{1}{2}f^\dagger_i\mathbf{\sigma}f_i$) mean-field study.

Let $H_{MF}=\sum_{<ij>}(f^\dagger_i\chi_{ij}f_j+f^\dagger_i\eta_{ij}f_j^\dagger+H.c.)$ be the resulting mean-field Hamiltonian, where $(\chi_{ij},\eta_{ij})$ is the mean-field ansatz. And let $\psi_{1,2}$ represent two exact eigenstates of $H_{MF}$ with energies $E_{1,2}(E_1>E_2)$, say $H_{MF}\psi_{1,2}=E_{1,2}\psi_{1,2}$.

Now we can construct the physical spin states $\phi_{1,2}$ by applying the projective operator $P=\prod_i(2\hat{n}_i-\hat{n}_i^2)$(Note that $P\neq \prod _i(1-\hat{n}_{i\uparrow}\hat{n}_{i\downarrow})$) to $\psi_{1,2}$(where $\hat{n}_i=f^\dagger_{i\uparrow}f_{i\uparrow}+f^\dagger_{i\downarrow}f_{i\downarrow}$), say $\phi_{1,2}=P\psi_{1,2}$, and generally we don't expect that $\phi_{1,2}$ are the exact eigenstates of the original spin Hamiltonian $H$.

My question is: $\frac{\left \langle \phi_1 \mid H \mid \phi_1 \right \rangle}{\left \langle \phi_1 \mid \phi_1 \right \rangle}>\frac{\left \langle \phi_2 \mid H \mid \phi_2 \right \rangle}{\left \langle \phi_2 \mid \phi_2 \right \rangle}$ ? If it's true, then how to prove it rigorously ? Thanks a lot.

This post imported from StackExchange Physics at 2014-03-09 08:43 (UCT), posted by SE-user K-boy

asked Aug 31, 2013
retagged Mar 25, 2014
Too much quantities are not defined precisely, and no reference is given, so you had better to completely rewrite your question, or to ask a new question.

This post imported from StackExchange Physics at 2014-03-09 08:43 (UCT), posted by SE-user Trimok
@ Trimok, I added some details to my post.

This post imported from StackExchange Physics at 2014-03-09 08:43 (UCT), posted by SE-user K-boy

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