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This program describes a common model for topological insulators without spin interaction. To begin with, a topological insulator differs from a common insulator with the existence of edge states in the normally open band gap, thus the possibility for electric current. Such points exist always in pairs (due to the Fermion double theorem) and they are time reversal symmetry related. Edge states are associated with time reversal breaking points - they form so called dirac cones. Contrary to the Hall effect no perpendicular magnetic field is needed to produce those edge channels. The band gap is determined by the mass term in the hamiltonian, therefore time reversal symmetry is conserved if that mass term vanishes.

The simplest insulator possesses two bands, one above and one below the band gap. Such an insulator can generically be described as a two-level system, which corresponds to a two-dimensional Hilbert space at each point of the Brillouin torus, on which acts a Bloch Hamiltonian continuously defined on the Brillouin torus. Hence H(k) can be written as a 2 × 2 Hermitian matrix (written in terms of the Pauli matrices - SU(2) algebra), parameterized by the real functions :

It is straightforward to obtain the eigenvalues:

Assuming h_0=0, which means the system remains insulating. The filled band of the two-bands insulator is described by a map that assigns a filled eigenvector to each point of the Brillouin torus: it defines a one-dimensional complex vector bundle on the torus. When this vector bundle is trivial, this map can be chosen to be continuous on the whole Brillouin torus: this corresponds to the standard situation where a choice of phase for the Bloch eigenstate at a given point k_0 of the Brillouin torus can be continuously extrapolated to the whole torus. When it is not trivial, there is an obstruction to do so.