Dynamical Symmetry Indicators for Floquet Crystals

TL;DR

This paper develops a comprehensive and efficient theoretical framework for classifying and identifying dynamical topological phases in Floquet crystals with crystalline symmetries, revealing new higher-order and 3+1D phases.

Contribution

It introduces quotient winding data and dynamical symmetry indicators for all crystalline symmetry groups in Floquet systems, enabling systematic classification and discovery of new topological phases.

Findings

Computed all elementary DSI sets for all plane groups.

Discovered a new 3+1D anomalous Floquet second-order topological insulator.

Identified a large variety of nontrivial Floquet topological classifications.

Abstract

Various exotic topological phases of Floquet systems have been shown to arise from crystalline symmetries. Yet, a general theory for Floquet topology that is applicable to all crystalline symmetry groups is still in need. In this work, we propose such a theory for (effectively) non-interacting Floquet crystals. We first introduce quotient winding data to classify the dynamics of the Floquet crystals with equivalent symmetry data, and then construct dynamical symmetry indicators (DSIs) to sufficiently indicate the "inherently dynamical" Floquet crystals. The DSI and quotient winding data, as well as the symmetry data, are all computationally efficient since they only involve a small number of Bloch momenta. We demonstrate the high efficiency by computing all elementary DSI sets for all spinless and spinful plane groups using the mathematical theory of monoid, and find a large number of different nontrivial classifications, which contain both first-order and higher-order 2+1D anomalous Floquet topological phases. Using the framework, we further find a new 3+1D anomalous Floquet second-order topological insulator (AFSOTI) phase with anomalous chiral hinge modes.