Exact diagonalization by Bogoliubov transformation


by arojo
Tags: bogoliubov, diagonalization, exact, transformation
arojo
arojo is offline
#1
Dec6-13, 04:31 AM
P: 16
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
Phys.Org News Partner Physics news on Phys.org
Chameleon crystals could enable active camouflage (w/ video)
Atomic switcheroo explains origins of thin-film solar cell mystery
X-ray laser experiment explores how specially shocked material gets stronger
DrDu
DrDu is offline
#2
Dec6-13, 07:57 AM
Sci Advisor
P: 3,377
Diagonalizing a 4x4 matrix can be done analytically, as the eigenvalues result as solutions of a fourth order polynomial which may be of special form. Have you tried?
arojo
arojo is offline
#3
Dec6-13, 09:11 AM
P: 16
Hi DrDu,

Actually I started by doing precisely that, but I got a messy result. Which certainly is analytical but hard "to read", at least from the point of view of getting an idea of what is going on without doing the numerics.
Actually I should rephrase my question as is there any elegant representation or expression of the diagonalization as for example the one obtained in BCS Superconductivity?
Thanks


Register to reply

Related Discussions
Confused about Bogoliubov transformation Atomic, Solid State, Comp. Physics 1
Bogoliubov transformation confusion Quantum Physics 0
Bogoliubov transformation Quantum Physics 11
Bogoliubov transformation 3-mode Atomic, Solid State, Comp. Physics 5
What is the Physical meaning of Bogoliubov transformation Atomic, Solid State, Comp. Physics 22