Quantization of the theory of topological insulators

Topological insulators are the solid bodies having a broad energy gap stable under small deformations. It motivates the usage of topological methods for their investigation. A key role in the theory of solid is played by their symmetry groups. It was Kitaev who has pointed out the relation between the symmetries of solid bodies and Clifford algebras. Following this idea the quantization of topological insulators should reduce to the theory of irreducible representations of Clifford algebras. The next important step was done by Kennedy and Zirnbauer who introduced the notion of pseudosymmetries. While the algebra of observables of topological insulators is formed by the Hamiltonians, satisfying the commutation relations with symmetry operators, the quantum observables are given by complex structures on the Nambu space, satisfying the anticommutation relations with pseudosymmetries. This correspondence determines the quantization of topological insulators.
The study is supported financially by Russian Scientific Foundation (grant N 24-11-00196).