# Does the Fermi surface make sense for "Fermi liquids" with non-uniform charge density?

For a Fermi liquid, the Fermi momentum is determined by the singularity of the Green's function at $\omega=0$, i.e., $G(\omega=0,{\bf k}={\bf k}_F)\to\infty$.

Suppose due to an external field or disorder, the charge density (or the chemical potential) is not uniform, i.e., it depends on the position. Now the system is not translational invariant, so the momentum is not a good quantum number, and we only have the Green's function in position space $G(\omega,{\bf x})$. Does the Fermi surface still make sense? is there a local Fermi surface, and how to define it?

There are actually two different questions here:

1) Does the concept of a Fermi-liquid extend to non translationally invariant systems?

2) If I have some kind of "slowly varying" perturbation to my electronic system, it should still look locally like a translationally invariant Fermi-liquid and hence have a well defined Fermi-surface locally. How do I extract the local parameters of my local Fermi-surface, locally speaking?

The answer to question one is yes, the Fermi-liquid concept extends. In the sense that the low energy theory is still a theory of weakly interacting particle-hole pairs and Cooper pairs. In the presence of disorder there is, as you say, no conserved momentum, but this means only that the pairs obey a diffusion equation instead of a wave equation. In another sense if I turn disorder on in 3-d Fermi-liquid there is no phase transition until I reach a critical disorder.

Question 2 is a more technical thing. In a Fermi-liquid things oscillate at the Fermi-wavelength, like Friedel osciallations, and the existence of these oscillations is a signature of the Fermi-surface. If we put a smooth external perturbation on our Fermi-liquid we expect things to oscillate at the "Fermi wavelength" where the Fermi wavelength itself is slowly changing with position. Whenever we have a wave with slowlying varying frequency we should Wigner transform. So define a Green's function $G(x,x',\omega)$ where I put an electron in at position $x$ and take it out at $x'$. Define the new function:

$$H(k,\omega;R) = \int dr \exp(ik\cdot r)G (R+r/2,R-r/2,\omega)$$

The function $H(k,\omega;R)$ is roughly what "$G(k,\omega)$ looks like near $R$". If everything is translationally invariant then $H$ reduces to the regular Green's function. The "local Fermi surface at $R$" can be extracted from the structure of $H(k,\omega;R)$ as in $G$ (although the discontinuity will only be approximate). You can write down equations of motion/ Dyson equation for $H$ just like you can for $G$ as long as you keep in mind that the with respect $R$ are much slower than the fermi wavelength.