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

Abstract

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 it 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 ${0,\phantom{\rule{0.16em}{0ex}}\ifmmode\pm\else\textpm\fi{}1\phantom{\rule{0.16em}{0ex}}\ifmmode\pm\else\textpm\fi{}2}$. 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 $\ifmmode\pm\else\textpm\fi{}2$.