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Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond


Nelson, Aleksandra; Neupert, Titus; Alexandradinata, A; Bzdušek, Tomáš (2022). Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond. Physical review B, 106(7):075124.

Abstract

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a 2π-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the bulk, three-dimensional Brillouin zone; because this change is reversed on the complementary section of the Brillouin zone, we refer to this new invariant as a returning Thouless pump (RTP). We find that the RTP is associated with anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary conditions with sharply terminated hoppings; more generally for arbitrarily terminated hoppings, surface states are characterized by Berry-Zak phases that are quantized to a rational multiple of 2π. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.

Abstract

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a 2π-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the bulk, three-dimensional Brillouin zone; because this change is reversed on the complementary section of the Brillouin zone, we refer to this new invariant as a returning Thouless pump (RTP). We find that the RTP is associated with anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary conditions with sharply terminated hoppings; more generally for arbitrarily terminated hoppings, surface states are characterized by Berry-Zak phases that are quantized to a rational multiple of 2π. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Physics Institute
Dewey Decimal Classification:530 Physics
Scopus Subject Areas:Physical Sciences > Electronic, Optical and Magnetic Materials
Physical Sciences > Condensed Matter Physics
Language:English
Date:10 August 2022
Deposited On:05 Sep 2022 15:09
Last Modified:27 Jun 2024 01:39
Publisher:American Physical Society
ISSN:2469-9950
OA Status:Green
Publisher DOI:https://doi.org/10.1103/physrevb.106.075124
Project Information:
  • : FunderSNSF
  • : Grant IDPP00P2_176877
  • : Project TitleTopological Phases: From New Fermions to Materials and Devices
  • : FunderSNSF
  • : Grant IDPZ00P2_185806
  • : Project TitleTopological band theory of driven and dissipative systems
  • : FunderHorizon 2020
  • : Grant ID
  • : Project Title
  • Content: Published Version
  • Licence: Creative Commons: Attribution 4.0 International (CC BY 4.0)