Entanglement spectrum classification of C-n-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 mth one-dimensional representation of the C-n point group. We find that for a space cut separating 1/pth of the system, the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues [1/p, 1 - 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. DOI:10.1103/PhysRevB.87.035119