$2+1$ dimensional Floquet systems and lattice fermions: Exact bulk spectral equivalence

TL;DR

This paper explores the spectral relationship between higher-dimensional Floquet insulators and static discrete-time fermion theories, demonstrating an exact bulk spectrum match in a specific model but noting differences in boundary behaviors.

Contribution

It provides an explicit example of a static discrete-time theory matching Floquet spectra in higher dimensions, extending previous one-dimensional results.

Findings

Exact bulk spectral match between Floquet and static models in a specific higher-dimensional example

Boundary spectra match up to finite-size corrections in a particular geometry

Differences in boundary behavior due to model anisotropy and dimensionality

Abstract

A connection has recently been proposed between periodically driven systems known as Floquet insulators in continuous time and static fermion theories in discrete time. This connection has been established in a $(1+1)$-dimensional free theory, where an explicit mapping between the spectra of a Floquet insulator and a discrete-time Dirac fermion theory has been formulated. Here we investigate the potential of static discrete-time theories to capture Floquet physics in higher dimensions, where so-called anomalous Floquet topological insulators can emerge that feature chiral edge states despite having bulk bands with zero Chern number. Starting from a particular model of an anomalous Floquet system, we provide an example of a static discrete-time theory whose bulk spectrum is an exact analytic match for the Floquet spectrum. The spectra with open boundary conditions in a particular strip geometry also match up to finite-size corrections. However, the models differ in several important respects. The discrete-time theory is spatially anisotropic, so that the spectra do not agree for all lattice terminations, e.g. other strip geometries or on half spaces. This difference can be attributed to the fact that the static discrete-time model is quasi-one-dimensional in nature and therefore has a different bulk-boundary correspondence than the Floquet model.