We know that metal, the metallic phase, in the nearly non-interacting (free) electron systems, we can fill the electron in the low energy low momentum space, and there will be a presence of Fermi surface with finite number of electrons (called finite electron density).
However, more fun things can happen if we **turn on some interactions** between the electrons, for example through: (1) Phonon (lattice vibration), (2) photon (through E&M), (3) emergent photon (through exotic behavior of condensed matter system), (3) Yukawa interactions through Higgs-like particles (such as polaron), (4) magenetic/spin excitations (magnons, etc), and so on.
=**Gapped or gapless**=
Moreover, we can ask whether the excited quasi-electron (quasi-particle) from the Fermi surface is:
**gapped or gapless.** Also whether the mediated gauge fields are **gapped or gapless.**
In short, there are many-possibilities!
Questions: What are some exotic examples of physics that happen in the presence of Fermi surface? How the Fermi surface plays any important role here?