Local and energy-resolved topological invariants for Floquet systems

TL;DR

This paper introduces a local, energy-resolved topological invariant based on the spectral localizer for Floquet systems, enabling detailed characterization of boundary modes and disorder effects in driven topological materials.

Contribution

It applies the spectral localizer to Floquet systems, providing a new method for unambiguously identifying and distinguishing $0$- and $\pi$-modes, even in disordered environments.

Findings

Successfully characterizes $0$- and $\pi$-boundary modes individually.

Demonstrates the method's effectiveness in disordered and phase transition scenarios.

Provides a spatially local topological invariant for Floquet systems.

Abstract

Periodically driven systems offer a perfect breeding ground for out-of-equilibrium engineering of topological boundary states at zero energy ($0$-mode), as well as finite energy ($\pi$-mode), with the latter having no static analog. The Floquet operator and the effective Floquet Hamiltonian, which encapsulate the stroboscopic features of the driven system, capture both spectral and localization properties of the $0$- and $\pi$-modes but sometimes fail to provide complete topological characterization, especially when $0$- and $\pi$-modes coexist. In this work, we utilize the spectral localizer, a powerful local probe that can provide numerically efficient, spatially local, and energy-resolved topological characterization. In particular, we apply the spectral localizer to the effective Floquet Hamiltonian for driven one- and two-dimensional topological systems with no or limited symmetries and are able to assign topological invariants, or local markers, that characterize the $0$- and the $\pi$-boundary modes individually and unambiguously. Due to the spatial resolution, we also demonstrate that the extracted topological invariants are suitable for studying driven disordered systems and can even capture disorder-induced phase transitions.