Calculations of Chern number: equivalence of real-space and twisted-boundary-condition formulae

TL;DR

This paper proves the equivalence between real-space and twisted-boundary-condition formulas for calculating the Chern number, a key topological invariant, through theoretical derivation and numerical validation in topological insulators.

Contribution

It establishes the theoretical equivalence between two different real-space formulas for the Chern number, enhancing understanding of topological invariants without translational symmetry.

Findings

Numerical confirmation for Chern insulator and quantum spin Hall insulator

Derivation of two real-space Chern number formulas

Establishment of equivalence between real-space and TBC formulas

Abstract

Chern number is a crucial invariant for characterizing topological feature of two-dimensional quantum systems. Real-space Chern number allows us to extract topological properties of systems without involving translational symmetry, and hence plays an important role in investigating topological systems with disorder or impurity. On the other hand, the twisted boundary condition (TBC) can also be used to define the Chern number in the absence of translational symmetry. Based on the perturbative nature of the TBC under appropriate gauges, we derive the two real-space formulae of Chern number (namely the non-commutative Chern number and the Bott index formula), which are numerically confirmed for the Chern insulator and the quantum spin Hall insulator. Our results not only establish the equivalence between the real-space and TBC formula of the Chern number, but also provide concrete and instructive examples for deriving the real-space topological invariant through the twisted boundary condition.