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

Mesoscale and Nanoscale Physics 2015-03-20 v1

Abstract

We study the single particle entanglement spectrum in 2D topological insulators which possess nn-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 ZnZ^n-index, (z1,...,zn)(z_1,...,z_n), where zmz_m is the number of occupied states that transform according to mm-th one-dimensional representation of the CnC_n point group. We find that the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues [1/n,11/n][1/n,1-1/n]. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the ZmZ_m quantum numbers. Furthermore, we show that in a homogeneous system, the ZnZ^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 nn-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.

Keywords

Cite

@article{arxiv.1208.4603,
  title  = {Entanglement Spectrum Classification of $C_n$-invariant Noninteracting Topological Insulators in Two Dimensions},
  author = {Chen Fang and Matthew J. Gilbert and B. Andrei Bernevig},
  journal= {arXiv preprint arXiv:1208.4603},
  year   = {2015}
}

Comments

14.2 pages for main text, 4.8 pages for Appendices, four figures and two tables