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

Aleksandra Nelson, Titus Neupert, A. Alexandradinata, and Tomáš Bzdušek
Phys. Rev. B 106, 075124 – Published 10 August 2022

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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  • Received 24 March 2022
  • Revised 9 July 2022
  • Accepted 22 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.075124

©2022 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Aleksandra Nelson1,*, Titus Neupert1,†, A. Alexandradinata2,‡, and Tomáš Bzdušek3,1,§

  • 1Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
  • 2Physics Department, University of California Santa Cruz, Santa Cruz, California 95064, USA
  • 3Condensed Matter Theory Group, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland

  • *anelson@physik.uzh.ch
  • titus.neupert@uzh.ch
  • aalexan6@ucsc.edu
  • §tomas.bzdusek@psi.ch

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Issue

Vol. 106, Iss. 7 — 15 August 2022

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