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  • Open Access

Geometric Floquet Theory

Paul M. Schindler* and Marin Bukov

  • *Contact author: psch@pks.mpg.de
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Phys. Rev. X 15, 031037 – Published 8 August, 2025

DOI: https://doi.org/10.1103/7l91-gw77

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)↦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 nonintegrable kicked Ising chain. We elucidate the geometric origin of inherently nonequilibrium effects, like the π-quasienergy splitting in discrete time crystals or π-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.

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Popular Summary

Periodic motion is central to many physical systems, from predator-prey dynamics to planetary orbits. Traditionally, we understand such systems using Floquet theory, which states that any periodically driven system can be described by a time-independent “Floquet Hamiltonian” in a suitable reference frame. While this framework has shed light on topics ranging from thermalization to topological materials, it faces two main challenges: It lacks a unique way to define the rotating frame, and calculating the effective Hamiltonian is often computationally difficult. In this work, we present an alternative formulation of Floquet theory rooted in quantum geometry to address both problems.

Our approach breaks down the system’s evolution into two parts: a geometric component and a dynamical one. By introducing the concept of a dynamical average-energy, we can sort the Floquet energy levels unambiguously and identify a Floquet ground state unambiguously. We also show that quantum geometry plays a key role in nonequilibrium phenomena, including anomalous Floquet topological insulators and time crystals, which we trace back to the geometric phase. Additionally, we find a duality between periodic driving and quantum control, allowing us to derive the Floquet theorem from the adiabatic theorem.

This geometric reformulation enables us to use powerful numerical tools to simulate Floquet systems more accurately and compare them with experiments. It also opens new directions for discovering and classifying nonequilibrium phases of matter based on their geometric and topological properties. Our work suggests a unified framework for understanding driven quantum systems.

Article Text

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Phys. Rev. X 15, 031037– Published 8 August, 2025
  • Received 10 February 2025
  • Revised 9 June 2025
  • Accepted 25 June 2025
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