Dynamic bulk-boundary correspondence for anomalous Floquet topology

TL;DR

This paper develops a systematic theoretical framework for understanding anomalous Floquet topological phases in periodically driven systems, revealing bulk-boundary relations across different symmetry classes and dimensions.

Contribution

It derives the first and second-order bulk-boundary correspondence for anomalous Floquet topology in various AZ symmetry classes using AHSS and dimensional hierarchy.

Findings

Established bulk-boundary correspondence for AFT in AZ classes BDI, D, DIII, AII.

Classified topological phases using a dimensional hierarchy and spectral sequence.

Provided analytical tools for studying out-of-equilibrium topological phenomena.

Abstract

Periodically driven systems with internal and spatial symmetries can exhibit a variety of anomalous boundary behaviors at both the zero and $\pi$ quasienergies despite the trivial bulk Floquet bands. These phenomena are called anomalous Floquet topology (AFT) as they are unconnected from their static counterpart, arising from the winding of the time evolution unitary rather than the bulk Floquet bands at the end of the driving period. In this paper, we systematically derive the first and inversion-symmetric second-order AFT bulk-boundary correspondence for Altland-Zirnbauer (AZ) classes BDI, D, DIII, AII. For each AZ class, we start a dimensional hierarchy with a parent dimension having $\mathbb{Z}$ classification, then use it as an interpolating map to classify the lower-dimensional descendants. From the Atiyah-Hirzebruch spectral sequence (AHSS), we identify the subspace that contains topological information and faithfully derive the AFT bulk-boundary correspondence for both the parent and descendants. Our theory provides analytic tools for out-of-equilibrium topological phenomena.