Fermionic thermal density matrix

Question

Usually to describe the density matrix of a system at finite temperature, we use the Euclidean path integral $$\rho[\psi_1,\psi_2] = \int _{\psi_1}^{\psi_2}\mathcal D \psi e^{-S_{E}[\psi]}, $$ where $S_E$ is the Euclidean action integrated for the Euclidean time $t_E\in [0,\beta]$. Then $\beta$ indicates the inverse temperature.

Question: If I am interested in describing a fermionic system, do I use the same procedure? If so, is it possible to see the existence of a Fermi-surface? Or do I have to input the Fermi surface by hand -- if so how can I do this? If it is too hard to see the existence of a Fermi surface in such a general case, is it possible to see it for a theory of free fermions? Any responses addressing some or all of these questions are welcome.

Note: I am only interested in seeing the existence of the Fermi surface using the path-integral; I am familiar with the usual derivation for Fermi gases/ liquids.

Answer 1

I'll take a statistical field theory approach with which I am more familiar with. But I think it's the same as yours. I'll omit any source term $J$ for simplicity.

The partition function is given by $$ Z = \mathrm{Tr} \left [ e^{-\beta \hat{H}} \right ] = \sum_\lambda \langle\lambda| e^{-\beta \hat{H}}|\lambda\rangle \\ = \int\mathcal{D}\phi\,\mathrm{e}^{-S_\mathrm{E}[\phi]},$$ where the "Euclidean action" $S_\mathrm{E}$ is related to the "Euclidean Lagrangian density" $\mathcal{L}_{\mathrm{E}}[\phi(x)]$ by $S_\mathrm{E} = \int_0^\beta \mathrm{d}\tau \int\mathrm{d}^3x \, \mathcal{L}_{\mathrm{E}}[\phi(x)]$.

Regarding the limits of integration, especially the imaginary time $\tau$:
If you were to employ periodic boundary conditions, you can set $\phi(\tau=0) = \phi(\beta)$ for bosons, but you would have to impose antiperiodic boundary conditions for fermions $\psi(\tau=0, \mathbf{x}) = -\psi(\tau=\beta, \mathbf{x}) $.

The result of a path integral is a number... and numbers "normally" commute. So, unknowingly and unintentionnaly, the above path integral is only valid for bosons.

In order to describe fermions with a path integral, we need numbers that anticommute. That is Grassmann numbers:

• $\eta \xi = -\xi \eta$,

• $\eta^2 = 0.$

Then, as shown for example here, you get a similar expression for the partition function:

$$ Z = \int\mathcal{D}\psi\,\mathcal{D}\bar{\psi}\,\mathrm{e}^{-S_\mathrm{E}[\psi, \bar{\psi}]},$$ where you have to use the antiperiodic boundary conditions outlined above.

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I don't know how to "show" the existence of the Fermi surface... but I'd say it should be automatic? If two numbers (particles) do not commute, then they cannot live in the same state and they need to "pile up". The level of this "pile" is the Fermi surface. I would assume this translates to the field perspective as well but I wouldn't know how to show it...