A real-space many-body marker for correlated ${\mathbb Z}_2$ topological insulators

TL;DR

This paper introduces a real-space operator based on the modern theory of polarization to distinguish between trivial and topological ${f Z}_2$ insulators in two dimensions, applicable to both non-interacting and interacting systems.

Contribution

It extends the position operator concept to two-dimensional systems and demonstrates its effectiveness for interacting models, enabling analysis of strongly-correlated topological insulators.

Findings

Successfully distinguishes trivial and topological phases in models

Applicable to interacting systems via Fock space formulation

Works for both non-interacting and small interacting clusters

Abstract

Taking the clue from the modern theory of polarization [R. Resta, Rev. Mod. Phys. {\bf 66}, 899 (1994)], we identify an operator to distinguish between ${\mathbb Z}_2$-even (trivial) and ${\mathbb Z}_2$-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [R. Resta and S. Sorella, Phys. Rev. Lett. {\bf 82}, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of non-interacting models, where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster, where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus allowing us to investigate generic interacting systems, such as strongly-correlated topological insulators.