Two-dimensional 2-band insulators breaking time reversal symmetry can present topological phases indexed by a topological invariant called the Chern number. We propose an efficient procedure to determine this topological index, which makes possible to conceive 2-band, tight-binding Hamiltonians with arbitrary Chern numbers. The technique is illustrated by a step by step construction of a model exhibiting five topological phases indexed by Chern numbers {0, ±1 ± 2}. On a finite cylindrical geometry, this insulator possesses up to two edge states which are characterized analytically. The model can be combined with its time reversal copy to form a quantum spin Hall insulator. It is shown that edge states in the latter can be destroyed by a time reversal invariant one-particle perturbation if the Chern number equals ±2. arXiv:1201.6613v3 [cond-mat.mes-hall]

The general properties of the phases with the icosahedral point group and long-range orientational order are considered. 6 Goldstone modes — 3 phonons and 3 phasons — are shown to exist. A model for the microscopic structure — a 6-D crystal — is proposed, and phason modes are discussed in this framework. Bravais lattice types are determined and some physical phenomena due to the peculiar AlMn dimension 6 are listed. Simple Landau-theory type arguments for advantages of icosahedral structure are put forward