A strange partition function

[zakk87]

While reading an article about superconductivity I found out a strange partition function which I don't know how to re-obtain. The partition function is given by:

$$Z=-\prod_{\omega,\mathbf{k}} (\omega^2 + E(\mathbf{k})^2)$$

where the sum over $\omega$ runs over Matsubara frequencies and $E(\mathbf{k})$ is the dispersion relation for the system.

I'm pretty sure that the procedure is quite general and does not depend upon the specific details of the the system and upon the analytical form of the dispersion relation.

P.S.: Anyone knows how to derive such a form for the partition function starting from the standard definition?

Thanks in advance.

Something very similar can be found in the first lines of paragraph 4.2 of this thesis: http://web.phys.ntnu.no/~mika/gjestland.pdf

 

[BigJDubs]

In this case the partition function is derived from the Hamiltonian of the system, which is written in terms of creation and annihilation operators. The Hamiltonian is then diagonalised through a Bogoliubov transformation and the result is written as a product of the energies of the quasiparticles, which gives the partition function.