Topology and solid-state physics (Review)

The topological aspect of the dynamics of electrons in a crystal (band electrons) and of crystal lattice vibrations (phonons) is discussed. The main features of the dynamics of conduction electrons in metals are connected with the shape of their Fermi surface, which is different from that for free electrons. It is demonstrated that the behavior of band electrons under the influence of external electric and magnetic fields depends strongly on the topology of the Fermi surface. Various examples of such a dependence (calculation of the periods of quantum oscillations, magnetic breakdown, features of the magnetoresistance, Bloch oscillations) are adduced and discussed. The features of the dynamics of phonons are manifested in singularities of the density of vibrational states (van Hove singularities), which are directly related to a change in the topology of the constant-frequency surfaces. The presence of a topological invariant that changes by a jump upon a change in topology of the constant-frequency surface is pointed out. The origin of the so-called phase transition of order two and a half is discussed.