TY - JOUR

T1 - Entanglement spectrum classification of Cn-invariant noninteracting topological insulators in two dimensions

AU - Fang, Chen

AU - Gilbert, Matthew J.

AU - Bernevig, B. Andrei

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/1/14

Y1 - 2013/1/14

N2 - 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/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 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.

AB - 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/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 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.

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U2 - 10.1103/PhysRevB.87.035119

DO - 10.1103/PhysRevB.87.035119

M3 - Article

AN - SCOPUS:84872918051

VL - 87

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 1098-0121

IS - 3

M1 - 035119

ER -