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

Understand

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 $[1/n,1-1/n]$.