Geometric Floquet theory

TL;DR

This paper develops a geometric Floquet theory derived from quantum geometry, providing new insights into quasienergy spectra, nonequilibrium effects, and transitionless driving in periodically driven quantum systems.

Contribution

It introduces a gauge-fixed geometric approach to Floquet theory, unifies concepts of quasienergy and dynamical evolution, and links nonequilibrium phenomena with geometric structures.

Findings

Identifies quasienergy folding as a broken gauge symmetry.

Defines a unique Floquet ground state via average-energy operator.

Demonstrates geometric origin of $ ext{pi}$-quasienergy splitting and edge modes.

Abstract

We derive Floquet theory from quantum geometry. We identify quasienergy folding as a consequence of a broken gauge group of the adiabatic gauge potential $U(1){\mapsto}\mathbb{Z}$. Fixing instead the gauge freedom using the parallel-transport gauge uniquely decomposes Floquet dynamics into a purely geometric and a purely dynamical evolution. The dynamical average-energy operator provides an unambiguous sorting of the quasienergy spectrum, identifying a Floquet ground state and suggesting a way to define the filling of Floquet-Bloch bands. We exemplify the features of geometric Floquet theory using an exactly solvable XY model and a non-integrable kicked Ising chain. We elucidate the geometric origin of inherently nonequilibrium effects, like the $\pi$-quasienergy splitting in discrete time crystals or $\pi$-edge modes in anomalous Floquet topological insulators. The spectrum of the average-energy operator is a susceptible indicator for both heating and spatiotemporal symmetry-breaking transitions. Last, we demonstrate that the periodic lab frame Hamiltonian generates transitionless counterdiabatic driving for Floquet eigenstates. This work directly bridges seemingly unrelated areas of nonequilibrium physics.