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

TL;DR

This paper introduces a method to design two-dimensional two-band Chern insulators with arbitrary topological indices, enabling the construction of tunable quantum anomalous Hall and quantum spin Hall phases with specific edge state properties.

Contribution

The authors develop an efficient procedure to determine and engineer Hamiltonians with any Chern number, expanding the design space for topological insulators.

Findings

Constructed a Chern insulator with Chern numbers from 0 to ±2.

Characterized edge states analytically in finite geometries.

Demonstrated how certain phases become trivial Z2 insulators.

Abstract

Two-dimensional 2-bands insulators breaking time reversal symmetry can present topological phases indexed by a topological invariant called the Chern number. Here we first propose an efficient procedure to determine this topological index. This tool allows in principle to conceive 2-bands Hamiltonians with arbitrary Chern numbers. We apply our methodology to gradually construct a quantum anomalous Hall insulator (Chern insulator) which can be tuned through five topological phases indexed by the Chern numbers {0,+/-1,+/-2}. On a cylindrical finite geometry, such insulator can therefore sustain up to two edge states which we characterize analytically. From this non-trivial Chern insulator and its time reversed copy, we build a quantum spin Hall insulator and show how the phases with a +/-2 Chern index yield trivial Z2 insulating phases.