Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index

TL;DR

This paper introduces a new class of three-dimensional topological phases protected by $ ext{PT}$ symmetry, characterized by a Pontryagin index, with unique linked nodal rings and potential realizations in metamaterials and ion traps.

Contribution

It identifies a novel $ ext{Z}$ invariant in 3D topological phases, linked to Pontryagin index and non-Abelian topology, expanding the understanding of multi-gap topological insulators and semimetals.

Findings

Identification of Pontryagin index as a bulk invariant

Characterization of linked nodal rings with non-Abelian charges

Proposal for realizing these phases in metamaterials and ion traps

Abstract

We report on a certain class of three-dimensional topological insulators and semimetals protected by spinless $\mathcal{P}\mathcal{T}$ symmetry, hosting an integer-valued bulk invariant. We show using homotopy arguments that these phases host multi-gap topology, providing a realization of a single $\mathbb{Z}$ invariant in three spatial dimensions that is distinct from the Hopf index. We identify this invariant with the Pontryagin index, which describes BPST instantons in particle physics contexts and corresponds to a 3-sphere winding number. We study naturally arising multi-gap linked nodal rings, topologically characterized by split-biquaternion charges, which can be removed by non-Abelian braiding of nodal rings, even without closing a gap. We additionally connect the describing winding number in terms of gauge-invariant combinations of non-Abelian Berry connection elements, indicating relations to Pontryagin characteristic class in four dimensions. These topological configurations are furthermore related to fully non-degenerate multi-gap phases that are characterized by a pair of winding numbers relating to two isoclinic rotations in the case of four bands and can be generalized to an arbitrary number of bands. From a physical perspective, we also analyze the edge states corresponding to this Pontryagin index as well as their dissolution subject to the gap-closing disorder. Finally, we elaborate on the realization of these novel non-Abelian phases, their edge states and linked nodal structures in acoustic metamaterials and trapped-ion experiments.