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

• A new method to calculate Berry phase in a two-level system for which the Hamiltonian is a real symmetric matrix.
• A convenient understanding about topological systems, such as AHE, topological semimetal and insulator.
• A simple way to determine the parameter range of the non-trivial Chern number in the phase diagram.

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 (1BZ1BZ) 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 −π2−π4sgn(B)[sgn(Δ−4B)+sgn(Δ)]. 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.