Real-space representation of winding number for one-dimensional chiral-symmetric topological insulator

TL;DR

This paper introduces a real-space method for calculating the winding number in one-dimensional chiral-symmetric topological insulators, accurately identifying topological phases even with disorder and linking it to the Bott index.

Contribution

The authors develop a real-space representation of the winding number that remains quantized under disorder and establish its equivalence to existing definitions, also connecting it to the Bott index.

Findings

The method reproduces quantized winding numbers in disordered systems.

It identifies topological phase transitions through ensemble averaging.

The real-space winding number can be expressed as a Bott index.

Abstract

The winding number has been widely used as an invariant for diagnosing topological phases in one-dimensional chiral-symmetric systems. We put forward a real-space representation for the winding number. Remarkably, our method reproduces an exactly quantized winding number even in the presence of disorders that break translation symmetry but preserve chiral symmetry. We prove that our real-space representation of the winding number, the winding number defined through the twisted boundary condition, and the real-space winding number derived previously in [Phys. Rev. Lett. 113, 046802 (2014)], are equivalent in the thermodynamic limit at half filling. Our method also works for the case of filling less than one half, where the winding number is not necessarily quantized. Around the disorder-induced topological phase transition, the real-space winding number has large fluctuations for different disordered samples, however, its average over an ensemble of disorder samples may well identify the topological phase transition. Besides, we show that our real-space winding number can be expressed as a Bott index, which has been used to represent the Chern number for two-dimensional systems.