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

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 Zⁿ index (z₁,...,zₙ), where zₘ is the number of occupied states that transform according to mth one-dimensional representation of the Cₙ 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,1-1/p]. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the Zⁿ index. Furthermore, we show that in a homogeneous system, the Zⁿ 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.