What's the real-space equivalent of the Fermi surface?

Question

So I'm familiar with the concept of Fermi surface in momentum space and all that. But if everything in the momentum space can be obtained by Fourier transforming something in the real space (e.g. crystal lattice <-> Brillouin zone; point-like electron <-> plane wave), how should I think about the Fermi surface in the real space?

Meaning, if the Fermi surface is a set of crystal momentum k with the same Fermi energy, in real space it should be a collection of electron's positions with the same something?

Answer 1

In the crystal lattice, free electrons propagate with different wavelengths. In a given crystal similar wavelengths tend to be either all occupied or all empty. The Fermi surface physically corresponds to the cut-off wavelengths were states go from occupied to unoccupied or vice-versa. So in physical space it is crystal-wide (since each electron state of given wavelength extend through the whole crystal), though it is a sharp surface in momentum space.

Ashcroft and Mermin has a good discussion for fermi surfaces with many useful example. In momentum space they can look rather exotic because it is not necessarily the higher wavelength crystal-wide states that are empty.