Entanglement Spectrum Classification of $C_n$-invariant Noninteracting Topological Insulators in Two Dimensions

TL;DR

This paper investigates how the entanglement spectrum in 2D topological insulators with $C_n$ symmetry reveals topological invariants, identifying protected in-gap states characterized by a $Z^n$-index and their robustness under disorder.

Contribution

It introduces a $Z^n$-index based on symmetry representations to classify topological phases via entanglement spectra in $C_n$-symmetric insulators, including disordered cases.

Findings

Protected in-gap states are determined by the $Z^n$-index.

The number of in-gap states is robust under symmetry-preserving disorder.

Entanglement spectrum features in-gap states within $[1/n,1-1/n]$ interval.

Abstract

We study the single particle entanglement spectrum in 2D topological insulators which possess $n$-fold rotation symmetry. By defining a series of special choices of subsystems on which the entanglement is calculated, or real space cuts, we find that the number of protected in-gap states for each type of these real space cuts is a quantum number indexing (if any) non-trivial topology in these insulators. We explicitly show the number of protected in-gap states is determined by a $Z^n$-index, $(z_1,...,z_n)$, where $z_m$ is the number of occupied states that transform according to $m$-th one-dimensional representation of the $C_n$ point group. We find that the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues $[1/n,1-1/n]$. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the $Z_m$ quantum numbers. Furthermore, we show that in a homogeneous system, the $Z^n$ index can be determined through an evaluation of the eigenvalues of point group symmetry operators at all high-symmetry points in the Brillouin zone. When disordered $n$-fold rotationally symmetric systems are considered, we find that the number of protected in-gap states is identical to that in the clean limit as long as the disorder preserves the underlying point group symmetry and does not close the bulk insulating gap.