Regularized universal topological local markers for Dirac systems

TL;DR

This paper introduces regularized universal topological local markers for Dirac systems that eliminate boundary irregularities, connect to known topological indices, and serve as stable indicators for disorder-induced phase transitions.

Contribution

It develops a regularized local marker compatible with periodic boundary conditions, linking it explicitly to established topological invariants and enhancing its stability in disordered systems.

Findings

Eliminates boundary irregularities in topological markers.

Establishes explicit connections to known topological indices.

Detects topological phase transitions in disordered systems.

Abstract

Local markers provide an efficient and powerful characterization of topological features of many systems, especially when the translation symmetry is broken. Recently, a universal topological local marker applicable in different symmetry classes of topological systems is proposed. However, it suffers from irregular behaviors at the boundary and its connection to other topological indexes remains elusive. In this work, we construct regularized universal topological local markers that apply to Dirac systems by utilizing position operators that are compatible with periodic boundary conditions. The regularized local markers eliminate the obstructive boundary irregularities successfully, and give rise to the desired global topological invariants such as the Chern number consistently when integrated over all the lattice sites. Furthermore, the regularized form allows us to establish an explicit connection between the local markers and some other known topological indices in two dimensions. For instance, it turns out to be equivalent to the Bott index in classes A, D, and C, and equivalent to the spin Chern number in classes DIII and AII. We further examine the utility and stability of this new marker in disordered scenarios. We find that its variance shows peaks at the phase boundaries, which promotes it as a useful indicator for detecting disorder-induced topological phase transitions.