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arXiv:1910.10180 · Physical review. B./Physical review. B

Identifying Cn -symmetric higher-order topology and fractional corner charge using entanglement spectra

Penghao Zhu, Kieran Loehr, Taylor L. Hughes

Published Mar 20, 2020• 19 citations
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Abstract

We study the entanglement spectrum (ES) of two-dimensional ${C}_{n}$-symmetric second-order topological insulators (TIs). We show that some characteristic higher-order topological observables, e.g., the filling anomaly and its associated fractional corner charge, can be determined from the ES of atomic and fragile TIs. By constructing the relationship between the configuration of Wannier orbitals and the number of protected in-gap states in the ES for different symmetric cuts in real space, we express the fractional corner charge in terms of the number of protected in-gap states of the ES. We show that our formula is robust in the presence of electron-electron interactions as long as the interactions preserve ${C}_{n}$ rotation symmetry and charge-conservation symmetry. Moreover, we discuss the possible signatures of higher-order topology in the many-body ES. Our methods allow the identification of some classes of higher-order topology without requiring the usage of nested Wilson loops or nested entanglement spectra.

DOI
10.1103/physrevb.101.115140

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We study the entanglement spectrum (ES) of two-dimensional Cn{C}_{n}Cn​-symmetric second-order topological insulators (TIs). We show that some characteristic higher-order topological observables, e.g., the filling anomaly and its associated fractional corner charge, can be determined from the ES of atomic and fragile TIs. By constructing the relationship between the configuration of Wannier orbitals and the number of protected in-gap states in the ES for different symmetric cuts in real space, we express the fractional corner charge in terms of the number of protected in-gap states of the ES. We show that our formula is robust in the presence of electron-electron interactions as long as the interactions preserve Cn{C}_{n}Cn​ rotation symmetry and charge-conservation symmetry. Moreover, we discuss the possible signatures of higher-order topology in the many-body ES. Our methods allow the identification of some classes of higher-order topology without requiring the usage of nested Wilson loops or nested entanglement spectra.
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Many-body higher-order topological invariant for CnC_nCn​-symmetric insulators
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa, we propose a well-defined many-body bulk invariant for CnC_nCn​ symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of CnC_nCn​ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as e/ne/ne/n with CnC_nCn​ symmetry, leading to a Zn\mathbb{Z}_nZn​ classification for higher-order topological insulators in two dimensions.
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We study the single-particle entanglement spectrum in 2D topological insulators which possess nnn-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{Z}^{n}Zn index (z1,...,zn)({z}_{1},...,{z}_{n})(z1​,...,zn​), where zm{z}_{m}zm​ is the number of occupied states that transform according to mmmth one-dimensional representation of the Cn{C}_{n}Cn​ point group. We find that for a space cut separating 1/p1/p1/pth of the system, the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues $[1/p,1\ensuremath{-}1/p]$. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the Zn{Z}^{n}Zn index. Furthermore, we show that in a homogeneous system, the Zn{Z}^{n}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 nnn-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.
Robust fractional charge localized at disclination defects has recently been found as a topological response in C6{C}_{6}C6​-symmetric 2D topological crystalline insulators (TCIs). In this article, we thoroughly investigate the fractional charge on disclinations in Cn{C}_{n}Cn​-symmetric TCIs, with or without time-reversal symmetry, and including spinless and spin-12\frac{1}{2}21​ cases. We compute the fractional disclination charges from the Wannier representations in real space and use band representation theory to construct topological indices of the fractional disclination charge for all 2D TCIs that admit a (generalized) Wannier representation. We find the disclination charge is fractionalized in units of en\frac{e}{n}ne​ for Cn{C}_{n}Cn​-symmetric TCIs, and for spin-12\frac{1}{2}21​ TCIs, with additional time-reversal symmetry, the disclination charge is fractionalized in units of 2en\frac{2e}{n}n2e​. We furthermore prove that with electron-electron interactions that preserve the Cn{C}_{n}Cn​ symmetry and many-body bulk gap, though we can deform a TCI into another which is topologically distinct in the free fermion case, the fractional disclination charge determined by our topological indices will not change in this process. Moreover, we use an algebraic technique to generalize the indices for TCIs with nonzero Chern numbers, where a Wannier representation is not applicable. With the inclusion of the Chern number, our generalized fractional disclination indices apply for all Cn{C}_{n}Cn​-symmetric TCIs. Finally, we briefly discuss the connection between the Chern number dependence of our generalized indices and the Wen-Zee term.
Two-dimensional higher-order topological insulators can display a number of exotic phenomena, such as half-integer charges localized at both corners and disclination defects. In this paper, we analyze these phenomena, focusing on the paradigmatic example of the quadrupole insulator with C4{C}_{4}C4​ rotation symmetry, and present a topological field theory description of the mixed geometry-charge responses. Our theory provides a unified description of the corner and disclination charges in terms of a physical geometry (which encodes disclinations), and an effective geometry (which encodes corners). We extend this analysis to interacting systems, and predict the response of fractional quadrupole insulators, which exhibit charge e/2(2k+1)e/2(2k+1)e/2(2k+1) bound to corners and disclinations.