Topological phases in the theory of solid states

This paper is devoted to the theory of topological phases — a new and actively developing direction in solid state physics. The topological phases are defined in the following way. Denote by $G$ the symmetry group and consider the set $\text{Ham}_G$ of classes of homotopy equivalent $G$-symmetric Hamiltonians. We assume that they have the energy gap stable under small deformations which makes it reasonable to use the topological methods for their study. It is possible to introduce on $\text{Ham}_G$ a natural stacking operation such that $\text{Ham}_G$, provided with this operation, becomes an Abelian monoid (i.e. an Abelian semigroup with the neutral element). The group of invertible elements of this monoid is precisely the topological phase. The initial ideas, lying in the base of the theory of topological phases, were formulated by Alexei Kitaev in his talks. It turns out that the family $(F_d)$ of $d$-dimensional topological phases forms an $\Omega$-spectrum. In other words, it has the property that the loop space $\Omega F_{d+1}$ is homotopy equivalent to the space $F_d$. This fact opens a way to wide use of algebraic topology methods for the study of topological phases. More concretely, one can associate with any $\Omega$-spectrum the generalized cohomology theory, determined by the functor $h^d$, which assigns to the topological space $X$ the set $[X,F_d]$ of classes of homotopy equivalent maps $X\to F_d$.