- #1
arojo
- 16
- 0
Hello all,
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
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