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

TL;DR

This paper introduces a new topological invariant called the returning Thouless pump (RTP) that classifies three-dimensional crystalline insulators with rotational symmetry, revealing delicate topological phases with unique surface states.

Contribution

It extends Pontrjagin's classification by incorporating rotational symmetry, defining the RTP invariant, and demonstrating its role in symmetry-protected delicate topology beyond two-band models.

Findings

RTP quantifies a $2\pi$-change in Berry-Zak phase between rotation-invariant lines.

Surface states exhibit anomalous angular momentum and quantized Berry-Zak phases.

RTP is a delicate invariant that can be generalized to multi-band Hamiltonians.

Abstract

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a $2\pi$-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 to anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary condition 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 $2\pi$. 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.