Energy spectrum of a metal and its singularities

The Fermi surface and the $\mathbf{p}$-space region adjacent to it shape the spectrum of the elementary excitations of a metal, the fermions (electrons and holes), as well as the bosons (phonons). Electron-phonon interaction renormalizes the dispersion laws of the elementary excitations. Their lifetime therefore becomes finite and the dependence of the energy on the quasimomentum has singularities. The features of these singularities are intimately related to the local geometry of the Fermi surface (to its shape, curvature, presence or absence of lines of parabolic points); this distinguishes them from other singularities (e.g., those due to phonon-phonon interaction). A unique role is played by the singularities, due to parabolic points on the Fermi surface, of the sound velocity as a function of the propagation direction, since these singularities are produced by electrons that have an infinite lifetime in a perfect crystal. The results cited formulate the general premises concerning the elementary-excitation spectrum of a metal and continue in this sense the semi-phenomenological approach developed by I. M. Lifshitz and his school.