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arXiv:2304.06974 · Physical review. B./Physical review. B

Bulk-edge correspondence of Stiefel-Whitney and Euler insulators through the entanglement spectrum and cutting procedure

Ryo Takahashi, Tomoki Ozawa

Published Aug 11, 2023• 13 citations
arXivHTMLPDF

Abstract

We propose an unconventional bulk-edge correspondence for two-dimensional Stiefel-Whitney insulators and Euler insulators, which are topological insulators protected by the $PT$ symmetry. We find that, although the energy spectrum under the open boundary condition is generally gapped, the entanglement spectrum is gapless when the Stiefel-Whitney or Euler class is nonzero. The robustness of the gapless spectrum for Stiefel-Whitney insulator can be understood through an emergent antiunitary particle-hole symmetry. For the Euler insulators, we propose a conjecture, which is supported by our numerical calculation, that the Euler class is equal to the number of crossings in the entanglement spectrum, taking into account the degree of the crossings. We also discuss that these crossings of the entanglement spectrum are related to the gap-closing points in the cutting procedure, which is the energy spectrum as the magnitude of the boundary hopping is varied.

DOI
10.1103/physrevb.108.075129

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We propose an unconventional bulk-edge correspondence for two-dimensional Stiefel-Whitney insulators and Euler insulators, which are topological insulators protected by the PTPTPT symmetry. We find that, although the energy spectrum under the open boundary condition is generally gapped, the entanglement spectrum is gapless when the Stiefel-Whitney or Euler class is nonzero. The robustness of the gapless spectrum for Stiefel-Whitney insulator can be understood through an emergent antiunitary particle-hole symmetry. For the Euler insulators, we propose a conjecture, which is supported by our numerical calculation, that the Euler class is equal to the number of crossings in the entanglement spectrum, taking into account the degree of the crossings. We also discuss that these crossings of the entanglement spectrum are related to the gap-closing points in the cutting procedure, which is the energy spectrum as the magnitude of the boundary hopping is varied.
In this study, we discuss a type of bulk-boundary correspondence, which holds for topological insulators and superconductors when the parity-time ( P T ) and/or parity-particle-hole ( P C ) symmetry are present. In these systems, even when the bulk topology is nontrivial, the edge spectrum is generally gapped, and thus the conventional bulk-boundary correspondence does not hold. We find that, instead of the edge spectrum, the single-particle entanglement spectrum becomes gapless when the bulk topology is nontrivial: i.e., the holds in P T - and/or P C -symmetric topological insulators and superconductors. After showing the correspondence using K -theoretic approach, we provide concrete models for each symmetry class up to three dimensions where nontrivial topology because of P T and/or P C is expected. An implication of our results is that, when the bulk topology under P T and/or P C symmetry is nontrivial, the noninteracting many-body entanglement spectrum is multiply degenerate in one dimension and is gapless in two or higher dimensions. Published by the American Physical Society 2024
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.
Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the PTPTPT or C2TC_2TC2​T symmetry in two dimensions, which go beyond the tenfold KKK-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.
Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.
We study a variety of questions related to entanglement in symmetry-protected phases. These phases are analogous to topological insulators in that they are short-range entangled states with symmetry-protected edge or surface states. We show that the now standard bulk-edge correspondence relating the entanglement spectrum to the gapless edge spectrum holds for these phases as well. We further consider the question of coupling these models to gauge fields which we show gives an alternative way to construct different kinds of long-range entangled states without symmetry. We also argue that this procedure yields models with perturbatively stable edge or surface states at a variety of interface types. The non-onsite nature of the edge symmetry plays an important role in our considerations.
In a recent work by Gong and Ueda [Phys. Rev. Lett. 121, 250601 (2018)], the classification of (1+1)-dimensional quench dynamics for the ten Altland-Zirnbauer classes is achieved, and entanglement-spectrum crossings of the time-dependent states for the topological classes (AIII, DIII, CII, BDI, and D) are discovered as a consequence of the bulk-edge correspondence. We note that their classification scheme for the (1+1)-dimensional bulk topology as well as the adoption of entanglement-spectrum crossings as a diagnosis of topological edge states focus on the limit that the spectrum of the postquench Hamiltonian is flat. When band dispersion of the postquench Hamiltonian is considered, even though the bulk topology is still stable, the entanglement-spectrum crossing at the edge might be unstable due to the fact that the band dispersion can be regarded as a ``symmetry-breaking disorder'' in the frequency domain. Such a symmetry-breaking disorder lowers the symmetry for the edge side, and the conventional bulk-edge correspondence could not be applied directly for the quench dynamics. We show that because of the reduction of symmetry by finite energy dispersion the gapless entanglement-spectrum crossing in the flatband limit in classes AIII, DIII, and CII is unstable and could be gapped without closing the bulk gap. The entanglement-spectrum crossing in classes BDI and D is still stable against energy dispersion. We show that the quench process for classes BDI and D can be understood as a Z2{\mathbb{Z}}_{2}Z2​ fermion parity pump, and the entanglement-spectrum crossing for this case is protected by the conservation of fermion parity.