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I want to prove the relation \eqref{eq:Metz_relation} that i found in this article.

\begin{equation} \left\langle\varPhi_0\right|\prod_{i} \hat{D}_i\left|\varPhi_0\right\rangle= \left\langle\varPhi_0\right|\prod_{i} \hat{n}_{i,\uparrow}\left|\varPhi_0\right\rangle \left\langle\varPhi_0\right|\prod_{i} \hat{n}_{i,\downarrow}\left|\varPhi_0\right\rangle \label{eq:Metz_relation}\tag{1} \end{equation}

where

  • $\hat{D}_i = \hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}$ , is the double occupation operator on site i
  • $\hat{n}_{i,\sigma} = \hat{c}^{\dagger}_{i,\sigma} \hat{c}_{i,\sigma}$, is the number operator
  • $\varPhi_0$, is a wave-function written in momentum space, for example a Slater determinant, on which i can apply the Wick's theorem

If i fix the number of the site in \eqref{eq:Metz_relation}, for example 2 lattice sites, i should find

\begin{equation} \left\langle\varPhi_0\right| \hat{D}_1 \hat{D}_2\left|\varPhi_0\right\rangle= \left\langle\varPhi_0\right|\hat{n}_{1,\uparrow}\hat{n}_{2,\uparrow}\left|\varPhi_0\right\rangle \left\langle\varPhi_0\right|\hat{n}_{1,\downarrow}\hat{n}_{1,\downarrow}\left|\varPhi_0\right\rangle \label{eq:Metz_relation_2}\tag{2} \end{equation}

I apply the Wick's theorem to the left member of \eqref{eq:Metz_relation_2}

\begin{align*} \left\langle\varPhi_0\right| \hat{D}_1 \hat{D}_2\left|\varPhi_0\right\rangle &= \left\langle\varPhi_0\right|\hat{n}_{1,\uparrow}\hat{n}_{1,\downarrow}\hat{n}_{2,\uparrow}\hat{n}_{2,\downarrow}\left|\varPhi_0\right\rangle \\ &= \left\langle\varPhi_0\right|\hat{n}_{1,\uparrow}\hat{n}_{1,\downarrow}\left|\varPhi_0\right\rangle \left\langle\varPhi_0\right|\hat{n}_{2,\uparrow}\hat{n}_{2,\downarrow}\left|\varPhi_0\right\rangle \\ &+ \left\langle\varPhi_0\right|\hat{n}_{1,\uparrow}\hat{n}_{2,\uparrow}\left|\varPhi_0\right\rangle \left\langle\varPhi_0\right|\hat{n}_{1,\downarrow}\hat{n}_{2,\downarrow}\left|\varPhi_0\right\rangle \\ &+ \left\langle\varPhi_0\right|\hat{n}_{1,\uparrow}\hat{n}_{2,\downarrow}\left|\varPhi_0\right\rangle \left\langle\varPhi_0\right|\hat{n}_{1,\downarrow}\hat{n}_{2,\uparrow}\left|\varPhi_0\right\rangle \end{align*}

if i use this relation

\begin{align} \left\langle\varPhi_0\right| \hat{n}_{i,\sigma}\hat{n}_{j,\sigma'} \left|\varPhi_0\right\rangle &= \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma} \hat{c}^{\dagger}_{j,\sigma'}\hat{c}_{j,\sigma'} \left|\varPhi_0\right\rangle \nonumber \\ &= \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma} \left|\varPhi_0\right\rangle \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{j,\sigma'}\hat{c}_{j,\sigma'} \left|\varPhi_0\right\rangle \nonumber \\ &- \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{i,\sigma}\hat{c}_{j,\sigma'} \left|\varPhi_0\right\rangle \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{j,\sigma'}\hat{c}_{i,\sigma} \left|\varPhi_0\right\rangle \delta_{\sigma,\sigma'} \label{eq:expval_nn} \tag{3} \end{align}

The expectation values of number operator products are

If $\sigma=\sigma'$

\begin{equation} \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{i,\sigma}\hat{c}_{j,\sigma} \left|\varPhi_0\right\rangle = \int \text{d}\mathbf{k} \, e^{-i\mathbf{k}\cdot \left(\mathbf{R}_{i}-\mathbf{R}_{j}\right)} n_{\mathbf{k},\sigma}^{0} = P_{i,j,\sigma} \label{eq:expval_nonull1}\tag{4} \end{equation}

Are the expectation values \eqref{eq:expval_nonull1}, with i=j, scalars?

while if $\sigma \neq \sigma'$ \begin{align} \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{i,\sigma}\hat{c}_{j,\sigma'} \left|\varPhi_0\right\rangle &= \left\langle\varPhi_0\right| \left( \frac{1}{\sqrt{V}}\sum_{k} e^{-i\mathbf{k}\cdot\mathbf{R}_{i}} \hat{c}^{\dagger}_{\mathbf{k},\sigma}\right) \left(\frac{1}{\sqrt{V}} \sum_{k'} e^{i\mathbf{k'}\cdot \mathbf{R}_{j}} \hat{c}_{\mathbf{k'},\sigma'} \right) \left|\varPhi_0\right\rangle \nonumber \\ &= \frac{1}{V} \sum_{k,k'} e^{-i\mathbf{k}\cdot \mathbf{R}_{i}} e^{i\mathbf{k'}\cdot \mathbf{R}_{j}} \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{\mathbf{k},\sigma} \hat{c}_{\mathbf{k'},\sigma'} \left|\varPhi_0\right\rangle \nonumber \\ &= \frac{1}{V} \sum_{k} e^{-i\mathbf{k} \cdot \left(\mathbf{R}_{i}-\mathbf{R}_{j}\right)} \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{\mathbf{k},\sigma} \hat{c}_{k,\sigma'} \left|\varPhi_0\right\rangle \nonumber \\ &+ \frac{1}{V} \sum_{k\neq k'} e^{-i\mathbf{k}\cdot \mathbf{R}_{i}} e^{i\mathbf{k'}\cdot \mathbf{R}_{j}} \left\langle\varPhi_0\right| \hat{c}^{\dagger}_{\mathbf{k},\sigma} \hat{c}_{\mathbf{k'},\sigma'} \left|\varPhi_0\right\rangle = 0 \label{eq:expval_ZERO}\tag{5} \end{align}

The ground state $\varPhi_0$ is written in momentum space, so the expectation values $\left\langle\varPhi_0\right| \hat{c}^{\dagger}_{\mathbf{k},\sigma} \hat{c}_{\mathbf{k},\sigma'}\left|\varPhi_0\right\rangle$ and $\left\langle\varPhi_0\right| \hat{c}^{\dagger}_{\mathbf{k},\sigma} \hat{c}_{\mathbf{k'},\sigma'}\left|\varPhi_0\right\rangle $ are scalar product of orthogonal states.

Using \eqref{eq:expval_nonull1} and \eqref{eq:expval_ZERO}, the expectation values,in \eqref{eq:expval_nn}, becomes

\begin{align*} \left\langle\varPhi_0\right|\hat{n}_{1,\uparrow}\hat{n}_{1,\downarrow}\left|\varPhi_0\right\rangle \left\langle\varPhi_0\right|\hat{n}_{2,\uparrow}\hat{n}_{2,\downarrow}\left|\varPhi_0\right\rangle + \left\langle\varPhi_0\right|\hat{n}_{1,\uparrow}\hat{n}_{2,\downarrow}\left|\varPhi_0\right\rangle \left\langle\varPhi_0\right|\hat{n}_{1,\downarrow}\hat{n}_{2,\uparrow}\left|\varPhi_0\right\rangle \\ &=2\, P_{1,1,\uparrow} P_{2,2,\uparrow} P_{1,1,\downarrow} P_{2,2,\downarrow} \end{align*}

so \eqref{eq:expval_nn}, becomes

\begin{align} \left\langle\varPhi_0\right| \hat{D}_1 \hat{D}_2 \left|\varPhi_0\right\rangle &= \left\langle\varPhi_0\right| \hat{n}_{1,\uparrow}\hat{n}_{2,\uparrow} \left|\varPhi_0\right\rangle \left\langle\varPhi_0\right| \hat{n}_{1,\downarrow}\hat{n}_{2,\downarrow} \left|\varPhi_0\right\rangle \\ &+ 2 P_{1,1,\uparrow} P_{2,2,\uparrow} P_{1,1,\downarrow} P_{2,2,\downarrow} \\ &= \left\langle\varPhi_0\right| \hat{n}_{1,\uparrow}\hat{n}_{2,\uparrow} \left|\varPhi_0\right\rangle \left\langle\varPhi_0\right| \hat{n}_{1,\downarrow}\hat{n}_{2,\downarrow} \left|\varPhi_0\right\rangle +\text{const.} \end{align}

maybe I'm not applying Wick's theorem for good, beacuse i find an extra term.

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