Floquet topological phases with symmetry in all dimensions

TL;DR

This paper explores symmetry-protected topological phases in dynamical systems, specifically Floquet drives, revealing new classes of topological order across all dimensions, including those outside traditional cohomology classifications.

Contribution

It constructs explicit Floquet drives for all cohomology classes and identifies novel dynamical topological phases beyond existing frameworks.

Findings

Constructed Floquet drives for each cohomology class in all dimensions.

Identified symmetry-protected Floquet phases outside cohomology.

Defined stable topological phases and introduced concepts of relative and absolute topological order.

Abstract

Dynamical systems may host a number of remarkable symmetry-protected phases that are qualitatively different from their static analogs. In this work, we consider the phase space of symmetry-respecting unitary evolutions in detail and identify several distinct classes of evolution that host novel dynamical order. Using ideas from group cohomology, we construct a set of interacting Floquet drives that generate dynamical symmetry-protected topological order for each nontrivial cohomology class in every dimension, illustrating our construction with explicit two-dimensional examples. We also identify a set of symmetry-protected Floquet drives that lie outside of the group cohomology construction, and a further class of symmetry-respecting topological drives which host chiral edge modes. We use these special drives to define a notion of phase (stable to a class of local perturbations in the bulk) and the concepts of relative and absolute topological order, which can be applied to many different classes of unitary evolutions. These include fully many-body localized unitary evolutions and time crystals.