I am reading this article.
The model hamiltonain of 2D square lattice for spinless fermions is written as:
$$H=H_{kin}+H_{int}=-\frac{J}{2}\sum_{<n,m>}c_n^\dagger c_m+\frac{V}{2}\sum_{<n,m>}n_nm_m$$ with J=hopping, V=interaction potential, $<n,m>$ nearest neighbor pairs and $n_n=c_n^\dagger c_n$=Number operator.
Baeriswyl wavefunction (BWF) can be written as:
$$|\psi_B>=N_B^{-1}\exp{(\tilde{\alpha}H_{kin})}|CDW>$$ and expression for CDW in k-space can be written as: $$|CDW>=\Pi_{k\epsilon RBZ}\frac{1}{\sqrt{2}}(c_k^\dagger +c_{k-Q}^\dagger)|0>$$ If we convert H into k-space also we will get something like this: $$H_{kin}=\sum_k\epsilon(k)c_k^\dagger c_k$$ $$H_{int}=-\frac{V}{N}\sum_{k,k',q}\epsilon(k)c_{k+q}^\dagger c_k c_{k'-q}^\dagger c_{k'}$$ To get final expression for $|\psi_B>$ (equ.3 in mentioned article) one have to apply $e^{\tilde{\alpha}H_{kin}}$ on $|CDW>$. I tried to solve this but couldn't succeed. Can anyone help me in this?

My Attempt

$$|\psi_B>=\Pi_{k\epsilon RBZ}\frac{N_B^{-1}}{\sqrt{2}} \exp{[\tilde{\alpha}{\sum_k\epsilon(k)c_k^\dagger c_k}]}(c_k^\dagger +c_{k-Q}^\dagger)|0>$$

$$|\psi_B>=\Pi_{k\epsilon RBZ}\frac{N_B^{-1}}{\sqrt{2}} [\exp{[\tilde{\alpha}{\sum_k\epsilon(k)c_k^\dagger c_k}]}c_k^\dagger |0> +\exp{[\tilde{\alpha}{\sum_k\epsilon(k)c_k^\dagger c_k}]}c_{k-Q}^\dagger)|0>]$$ In the article they are saying that this is equal to $$|\psi_B>=\Pi_{k\epsilon RBZ}\frac{N_B^{-1}}{\sqrt{2}} [\exp{[\tilde{\alpha}{\sum_k\epsilon(k)}]}c_k^\dagger |0> +\exp{[\tilde{\alpha}{\sum_k\epsilon(k)}]}c_{k-Q}^\dagger)|0>]$$ But how?


$H_{kin}$ can be written as: $$\begin{bmatrix}c_k^\dagger & c_{k-\pi}^\dagger\end{bmatrix}\begin{bmatrix}\epsilon(k) & 0 \\0 & \epsilon(k- \pi)\end{bmatrix}\begin{bmatrix}c_k \\ c_{k-\pi}\end{bmatrix}$$ and $\epsilon(k-\pi)=-2*t*\cos(k-\pi)=2*t*\cos k=-\epsilon(k)$ and at half-filling $Q=\pi$. As $c_k^\dagger$ and $c_{k-\pi}^\dagger$ are eigenstates of $H_{kin}$ so one can write $\exp(\alpha H_{kin})(c_k^\dagger+c_{k-\pi}^\dagger)=\exp(\alpha \epsilon(k))c_k^\dagger+\exp(-\alpha \epsilon(k))c_{k-\pi}^\dagger$

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