Mapping topological order in coordinate space

TL;DR

This paper introduces a method to spatially map the topological order of a 2D insulator using a topological marker in real space, applicable under various boundary conditions, validated through simulations.

Contribution

It presents a novel real-space topological marker for mapping the Chern number in 2D insulators, enabling local topological characterization.

Findings

The topological marker accurately maps Chern number variations.

The method works for both periodic and open boundary conditions.

Simulations confirm the validity of the topological mapping.

Abstract

The organization of the electrons in the ground state is classified by means of topological invariants, defined as global properties of the wavefunction. Here we address the Chern number of a two-dimensional insulator and we show that the corresponding topological order can be mapped by means of a "topological marker", defined in $\r$-space, and which may vary in different regions of the same sample. Notably, this applies equally well to periodic and open boundary conditions. Simulations over a model Hamiltonian validate our theory.