Tolopogical phases in solid state physics

Topological methods play an important role in the theory of solid states. They were successfully applied to the theory of insulators characterized by a wide energy gap stable under small deformations. The availability of such gap is also important for the theory of topological phases.
To define the topological phases denote by $G$ the symmetry group and consider the set $\text{Ham}_G$ of homotopy equivalence classes of $G$-symmetric Hamiltonians satisfying the energy gap condition. It is possible to introduce a natural stacking operation on this set making $\text{Ham}_G$ into Abelian monoid (i.e. Abelian semigroup with neutral element). The group of invertible elements of this monoid is precisely the topological phase.
It turns out that the family $(F_d)$ of $d$-dimensional topological phases forms an $\Omega$-spectrum. By this we mean a collection of topological spaces $F_d$ having the property that the loop space $\Omega F_{d+1}$ is homotopy equivalent to the space $F_d$. Every $\Omega$-spectrum generates a generalized cohomology theory determined by the functor $h^d$, associating with the topological space $X$ the set $[X,F_d]$ of homotopy equivalence classes of the maps $X\to F_d$. We give several examples of concrete physical systems which can be described in terms of generalized cohomology theories.
We also discuss relations between topological phases and K-theory. The K-functor is defined using the spectral flattening of Hamiltonians and can be computed in a series of important examples.