On Mathematical Aspects of the Theory of Topological Insulators

This review is devoted to one of the most interesting and actively developing fields in condensed matter physics—theory of topological insulators. Apart from its importance for theoretical physics, this theory enjoys numerous connections with modern mathematics, in particular, with topology and homotopy theory, Clifford algebras, K-theory and non-commutative geometry. From the physical point of view topological invariance is equivalent to adiabatic stability. Topological insulators are characterized by the broad energy gap, stable under small deformations, which motivates application of topological methods. A key role in the study of topological objects in the solid state physics is played by their symmetry groups. There are three main types of symmetries—time reversion symmetry, preservation of the number of particles (charge symmetry) and PH-symmetry (particle-hole symmetry). Based on the study of symmetry groups and representation theory of Clifford algebras Kitaev proposed a classification of topological objects in solid state physics. In this review we pay special attention to the topological insulators invariant under time reversion.