A Characterization of Topological Insulators: Chern Numbers for a Ground State Multiplet

TL;DR

This paper introduces a numerical method using non-Abelian Chern numbers to characterize topological insulators, especially in strongly fluctuating quantum systems, by analyzing ground state multiplets through boundary condition twists.

Contribution

It develops a framework for defining and computing Chern numbers for ground state multiplets in finite systems, applicable to topological insulators and fractional quantum Hall states.

Findings

Chern numbers vanish for symmetric two-dimensional XXZ-spin systems.

The method distinguishes topological phases via energy gaps in ground state multiplets.

Application to fractional quantum Hall states briefly discussed.

Abstract

We propose to use generic Chern numbers for a characterization of topological insulators. It is suitable for a numerical characterization of low dimensional quantum liquids where strong quantum fluctuations prevent from developing conventional orders. By twisting parameters of boundary conditions, the non-Abelian Chern number are defined for a few low lying states near the ground state in a finite system, which is a ground state multiplet with a possible (topological) degeneracy. We define the system as a topological insulator when energies of the multiplet are well separated from the above. Translational invariant twists up to a unitary equivalence are crutial to pick up only bulk properties without edge states. As a simple example, the setup is applied for a two-dimensional $XXZ$-spin system with an ising anisotropy where the ground state multiplet is composed of doubly almost degenerate states. It gives a vanishing Chern number due to a symmetry. Also Chern numbers for the generic fractional quantum Hall states are discussed shortly.