Entanglement spectrum classification ofCn-invariant noninteracting topological insulators in two dimensions

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) nontrivial topology in these insulators. We explicitly show that 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 for a space cut separating $1/p$th of the system, the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues $[1/p,1\ensuremath{-}1/p]$. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the ${Z}^{n}$ index. 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.