Topology and physics of solids

The topological aspect of the dynamics of electrons in crystals (band electrons) and vibrations of a crystal lattice (phonons) is discussed. The main peculiarities of such electrons are connected with the form of their isoenergetic surfaces which differ from those for free electrons. It is shown that in response to external electric and magnetic fields the behavior of the band electrons in metals at low temperatures depends strongly on the Fermi surface topology (the isoenergetic surfaces for the Fermi energy). Various examples of such a dependence are described. The peculiarities of the phonon dynamics create singularities of the vibration density of states (van Hove singularities), which are connected with the topology of isoenergetic surfaces. The nature of what is known as 'the second and a half'-kind phase transition is explained