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

الأصل · EN

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 Zⁿ-index, (z₁,...,zₙ), where zₘ is the number of occupied states that transform according to m-th one-dimensional representation of the Cₙ 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 Zₘ quantum numbers. 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.