Group theory and topology in solid state physics

Abstract

Firstly, various formal concepts of finite group theory are discussed with particular reference to their use in recent work on solid state theory. The pictorial viewpoint is stressed wherever possible throughout the discussion. Groups of operators are treated, including the anti-unitary operators involved in the magnetic groups. Next, the use of group theory in conjunction with matrix methods is discussed, and this leads to the treatment of equivalent operators. An account is given of several methods used in the many-electron problem, including a treatment of second quantization from a group theoretical viewpoint. The class sum operator approach is outlined as a means of linking the group theoretical approach and the traditional Dirac approach to quantum mechanics. Some topics from crystal field theory and electronic band theory are treated as illustrations of the general principles, and some recent work in these two fields is reviewed. The basic terminology of graph theory is then given, and several applications to solid state theory are treated. Some topics which involve the fixed points and indices of mappings are discussed, as are the similarities and differences between the theory of one-, two- and three-dimensional lattices. An account is given of the effect of periodic boundary conditions in reciprocal space on the dispersion curves for lattice vibrations, with particular reference to the theory of critical points in the density of states function. Finally, a survey is given of experimental methods which employ strong magnetic fields to investigate the topology of the Fermi surface in metals.