Geometric Floquet theory

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.

9 + 10 pages, 5 + 5 figures

References in corpus (67)

• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Experimental realisation of the topological Haldane model
• Universal High-Frequency Behavior of Periodically Driven Systems: from Dynamical Stabilization to Floquet Engineering
• Shortcuts to adiabaticity: concepts, methods, and applications
• Direct Measurement of the Zak phase in Topological Bloch Bands
• Floquet Engineering of Quantum Materials
• Topological Bands for Ultracold Atoms
• On the phase structure of driven quantum systems
• High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective
• Shortcut to adiabatic passage in two and three level atoms
• Discrete time crystals: rigidity, criticality, and realizations
• Long-time behavior of periodically driven isolated interacting lattice systems
• Shortcuts to adiabaticity by counter-diabatic driving
• Thermalization and prethermalization in isolated quantum systems: a theoretical overview
• Floquet-Magnus Theory and Generic Transient Dynamics in Periodically Driven Many-Body Quantum Systems
• Exponentially slow heating in periodically driven many-body systems
• Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems
• Effective Hamiltonians, prethermalization and slow energy absorption in periodically driven many-body systems
• Geometry and non-adiabatic response in quantum and classical systems
• Realization of an anomalous Floquet topological system with ultracold atoms
• Topological singularities and the general classification of Floquet-Bloch systems
• Pre-thermal phases of matter protected by time-translation symmetry
• Equilibration and Order in Quantum Floquet Matter
• An Aharonov-Bohm interferometer for determining Bloch band topology
• Many-body energy localization transition in periodically driven systems
• Floquet-engineering counterdiabatic protocols in quantum many-body systems
• Thouless pumping and topology
• Colloquium: Quantum and Classical Discrete Time Crystals
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• Generating shortcuts to adiabaticity in quantum and classical dynamics
• Adiabatic tracking of quantum many-body dynamics
• Adiabatic eigenstate deformations as a sensitive probe for quantum chaos
• Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model
• Floquet prethermalization and regimes of heating in a periodically driven, interacting quantum system
• Adiabatic Perturbation Theory and Geometry of Periodically-Driven Systems
• Quantum localization bounds Trotter errors in digital quantum simulation
• Heating and many-body resonances in a periodically driven two-band system
• Onset of Floquet Thermalisation
• Long-lived interacting phases of matter protected by multiple time-translation symmetries in quasiperiodically-driven systems
• Logarithmically Slow Relaxation in Quasi-Periodically Driven Random Spin Chains
• Anomalous Floquet Insulators
• From the adiabatic theorem of quantum mechanics to topological states of matter
• Counterdiabatic Optimised Local Driving
• Towards adiabatic quantum computing using compressed quantum circuits
• Random multipolar driving: tunably slow heating through spectral engineering
• Quantum and classical Floquet prethermalization
• Fast adiabatic evolution by oscillating initial Hamiltonians
• Nonadiabatic simulation study of photoisomerization of azobenzene: Detailed mechanism and load-resisting capacity
• Shortcuts to Adiabaticity in Krylov Space
• Driving protocol for a Floquet topological phase without static counterpart
• Almost strong 0, π edge modes in clean, interacting 1D Floquet systems
• Local readout and control of current and kinetic energy operators in optical lattices
• Many-Body Localization with Quasiperiodic Driving
• Optimal steering of matrix product states and quantum many-body scars
• Shortcuts to Adiabatic Classical Spin Dynamics Mimicking Quantum Annealing
• Efficient Paths for Local Counterdiabatic Driving
• Hierarchy of many-body invariants and quantized magnetization in anomalous Floquet insulators
• Counterdiabatic Driving for Periodically Driven Systems
• Floquet topological phase transitions in a kicked Haldane-Chern insulator
• Variational adiabatic transport of tensor networks
• Robust effective ground state in a nonintegrable Floquet quantum circuit
• Sharp detection of the onset of Floquet heating using eigenstate sensitivity
• Counterdiabatic Route to Entanglement Steering and Dynamical Freezing in the Floquet Lipkin-Meshkov-Glick Model
• The missing quantum number of the Floquet states
• The anomalous Floquet Anderson insulator in a continuously driven optical lattice
• Effective (Floquet) Lindblad generators from spectral unwinding

Cited by in corpus (9)

• Taming quantum systems: A tutorial for using shortcuts-to-adiabaticity, quantum optimal control, and reinforcement learning
• Walsh-Floquet Theory of Periodic Kick Drives
• Prethermalization, shadowing breakdown, and the absence of Trotterization transition in quantum circuits
• Reciprocal Floquet thermalization in one-dimensional Rydberg atom array
• Higher order Magnus expansions for driven two-level quantum dynamics
• Detuning the Floquet anomalous chiral spin liquid
• Non-abelian Geometric Quantum Energy Pump
• Finding periodic orbits in projected quantum many-body dynamics
• Trotter transition in BCS pairing dynamics