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I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by,

[tex]

H =

\begin{pmatrix}

\xi_\mathbf{k} & -\sigma U_1 & -U_2 & -U_2\\

-\sigma U_1 & \xi_{\mathbf{k}+(\pi,\pi)} & 0 & 0\\

- U_2 & 0 & \xi_{\mathbf{k}+(\pi/2,0)} & 0\\

- U_2 & 0 & 0 & \xi_{\mathbf{k}+(0,\pi/2)}

\end{pmatrix}

[/tex]

And my Nambu operator is given by,

[tex]

ψ_\mathbf{k} =

\begin{pmatrix}

c_{\mathbf{k},\sigma} \\

c_{\mathbf{k}+(\pi,\pi),\sigma} \\

c_{\mathbf{k}+(\pi/2,0),\sigma} \\

c_{\mathbf{k}+(0,\pi/2),\sigma}

\end{pmatrix}

[/tex]

I tried to diagonalized by making three Bogoliubov transformations, the first to diagonalize the upper right submatrix of H, and then the two others (a sort of nested transformations). But I get a lengthy result, what I would like to know if there is a smart transformation which allows me to write

[tex] H = A_1^\dagger A_2^\dagger A_3^\dagger D A_3 A_2 A_1 [/tex]

or simply

[tex] H = U^\dagger D U [/tex]

Or the only way is to use just brute force?

Thanks

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# Exact diagonalization by Bogoliubov transformation

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