A new method to calculate Berry phase in one-dimensional quantum anomalous Hall insulator

Based on the residue theorem and degenerate perturbation theory, we derive a new, simple and general formula for Berry phase calculation in a two-level system for which the Hamiltonian is a real symmetric matrix. The special torus topology possessed by the first Brillouin zone (1 BZ) of this kind of systems ensures the existence of a nonzero Berry phase. We verify the correctness of our formula on the Su-Schrieffer-Heeger (SSH) model. Then the Berry phase of one-dimensional quantum anomalous Hall insulator (1DQAHI) is calculated analytically by applying our method, the result being -pi/2 - pi/4 sgn(B)(sgn(Delta-4B)+sgn(Delta)]. Finally, illuminated by this idea, we investigate the Chern number in the two-dimensional case, and find a very simple way to determine the parameter range of the non-trivial Chern number in the phase diagram. (C) 2016 Elsevier B.V. All rights reserved.