Partial Tracing Techniques for Fermionic Partition Functions

[ledamage]

Hi there!

Up to now, I've been not so familiar with theoretical condensed matter physics but now I have to calculate a partition function of the type

$$Z = \mathrm{Tr}\,\mathrm{e}^{-\beta(a^\dagger a + a^\dagger b + ab^\dagger)}$$

where $a, a^\dagger, b, b^\dagger$ are fermionic annihilition/creation operators. I want to take only a partial trace over the $a$-particles. I've tried several things such as BHC and the Trotter product formula and evaluation of the exponential for certain parts of the Hamiltonian but I've obtained nothing which is actually feasible. I've had a look in several books about many-particle quantum theory but found nothing useful. Is this problem elementary? Any ideas or literature recommendations?

Thanks!

 

[weejee]

I recommend the path integral technique, which is introduced in many field theory books.

 

[ledamage]

Okay, I guessed as much. Thanks!

Edit: Maybe, yet still another question: this is a finite dimensional Fock space with four basis elements (the tensor products of one 0-particle and one 1-particle state for both a and b). Are there really no well-known methods to decompose density matrices like this and evaluate the trace within the second quantization formalism?

 

[genneth]

Why not just write down the possible states and do the trace? It's just a finite sum, as you've noticed.

 

[ledamage]

You're right if I wanted to do the complete trace. Then one could diagonalize the Hamiltonian and do the sum. But what if I just want to do the trace over the a-particles? Then diagonalizing would destroy the possibility of doing the partial trace easily, wouldn't it? And otherwise, I have to find a decomposition of the exponential because direct evaluation yields awkward sums which I couldn't do in closed form.