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Two-dimensional 2-bands insulators breaking time reversal symmetry canpresent topological phases indexed by a topological invariant called the Chernnumber. Here we first propose an efficient procedure to determine thistopological index. This tool allows in principle to conceive 2-bandsHamiltonians with arbitrary Chern numbers. We apply our methodology togradually construct a quantum anomalous Hall insulator (Chern insulator) whichcan be tuned through five topological phases indexed by the Chern numbers{0,+/-1,+/-2}. On a cylindrical finite geometry, such insulator can thereforesustain up to two edge states which we characterize analytically. From thisnon-trivial Chern insulator and its time reversed copy, we build a quantum spinHall insulator and show how the phases with a +/-2 Chern index yield trivial Z2insulating phases.

Geometrical engineering of a two-bands Chern insulator in two dimensions with arbitrary topological index