Computing the $\mathbb{Z}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems

TL;DR

This paper develops a new formulation for the $ ext{Z}_2$ topological invariant in two-dimensional insulators, applicable to both non-interacting and strongly-correlated systems, and applies it to the Kane-Mele model with interactions.

Contribution

It introduces a boundary-condition dependence approach to compute the $ ext{Z}_2$ invariant for strongly-correlated insulators, extending previous band theory methods.

Findings

Derived an integral expression for the $ ext{Z}_2$ invariant in correlated systems.

Proved equivalence of many-body and band-theoretic formulations.

Calculated the invariant for the Kane-Mele model with nonlocal interactions.

Abstract

We show that the two-dimensional $\mathbb{Z}_2$ invariant for time-reversal invariant insulators can be formulated in terms of the boundary-condition dependence of the ground state wavefunction for both non-interacting and strongly-correlated insulators. By introducing a family of quasi-single particle states associated to the many-body ground state of an insulator, we show that the $\mathbb{Z}_2$ invariant can be expressed as the integral of a certain Berry connection over half the space of boundary conditions, providing an alternative expression to the formulations that appear in [Lee et al., Phys. Rev. Lett. $\textbf{100}$, 186807 (2008)]. We show the equivalence of the different many-body formulations of the invariant, and show how they reduce to known band-theoretic results for Slater determinant ground states. Finally, we apply our results to analytically calculate the invariant for the Kane-Mele model with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. This rigorously establishes the topological nontriviality of the Kane-Mele model with HK interactions, and represents one of the few exact calculations of the $\mathbb{Z}_2$ invariant for a strongly-interacting system.