Floquet Thermalization

• Floquet thermalization is the late-time equilibration of periodically driven many-body systems where the stroboscopic evolution via the Floquet operator replaces static Hamiltonian control.
• It involves a multistage process with an initial transient, a prethermal plateau governed by an effective Hamiltonian, and eventual heating to an infinite-temperature state.
• Mechanisms such as high-frequency driving, emergent quasiconserved quantities, and localization phenomena shape and delay the thermalization pathway while diagnostics include quasienergy statistics, entanglement, and operator autocorrelations.

Searching arXiv for recent and foundational papers on Floquet thermalization. Found relevant arXiv papers spanning foundational, methodological, experimental, and recent perspectives on Floquet thermalization. Floquet thermalization is the late-time equilibration of an isolated periodically driven many-body system under the stroboscopic evolution generated by its Floquet operator, rather than by a static Hamiltonian. For a drive with period $T$, the basic object is $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$, with quasienergies defined modulo $2\pi/T$. In generic interacting settings without relevant conservation laws, the expected endpoint is heating to an infinite-temperature Floquet state, so local observables lose memory of the initial condition. At the same time, high-frequency driving, emergent quasiconserved quantities, symmetry constraints, localization phenomena, and geometry-induced bottlenecks can delay, reshape, or even obstruct that route to thermalization. The subject is therefore organized by the relation among quasienergy statistics, Floquet eigenstate structure, ETH, and the mechanisms that either enable or suppress heating (Liao et al., 2022, Regnault et al., 2015, Mukherjee et al., 2024).

1. Formal framework and meaning of thermalization

A Floquet system is defined by a time-periodic Hamiltonian $H(t+T)=H(t)$. Its stroboscopic dynamics is governed by $U_F$, whose eigenstates $\{|\Phi_\alpha\rangle\}$ satisfy

$$U_F|\Phi_\alpha\rangle=e^{-i\epsilon_\alpha T}|\Phi_\alpha\rangle.$$

Because quasienergies are defined on a circle, energy conservation is generically absent, and the natural thermal endpoint is not a Gibbs ensemble fixed by the microscopic Hamiltonian but the infinite-temperature state within the sector of exact conserved quantities. This is the sense in which Floquet thermalization differs from thermalization in static Hamiltonian systems (Russomanno et al., 2014, Dudinets et al., 2024).

That generic picture is clearest in models without local conservation laws. A disordered Floquet spin chain introduced as a clean numerical setting for the MBL transition was constructed specifically so that its thermal phase relaxes rapidly to the unconstrained maximum-entropy state, sharpening the distinction between thermal and localized behavior in finite-size exact diagonalization (Zhang et al., 2016). In more general interacting drives, recent subsystem-based analyses have emphasized that one must distinguish true asymptotic heating from effective late-time descriptions on accessible timescales: low frequencies drive the system toward $\beta\to 0$, intermediate frequencies can support prethermal plateaus, and high frequencies may admit a finite-$\beta$ effective description over the simulated window rather than a permanently nonheating phase (Lin et al., 1 Jul 2026).

This distinction between asymptotic fate and long-lived intermediate dynamics is central. Several works now treat Floquet thermalization as a multistage process: an initial transient, a prethermal regime described by an effective Hamiltonian, and a later heating stage in which residual nonperturbative processes or higher-order drive effects finally push the system toward infinite temperature (Fleckenstein et al., 2021, Mukherjee et al., 2024).

2. Floquet ETH, eigenstate statistics, and random-matrix structure

The most direct formulation of Floquet thermalization is in the Floquet eigenbasis. In this language, thermalization means that Floquet eigenstates satisfy a Floquet version of ETH: diagonal matrix elements of local observables take their infinite-temperature values, while off-diagonal matrix elements are parametrically small. For Floquet systems without conserved quantities, the ETH ansatz takes the form

$$O_{\mu\nu}=\bar O\,\delta_{\mu\nu}+N^{-1/2}f(\Delta E_{\mu\nu})R_{\mu\nu},\qquad \bar O=\frac{\mathrm{Tr}(O)}{N},$$

with no dependence on an average energy because quasienergies live on a circle and the thermal endpoint is infinite temperature (Liao et al., 2022).

A field-theoretic realization of this picture was given for a broad family of local Floquet random quantum circuits on arbitrary-dimensional square lattices of qudits. In the large-$$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$0 limit, the disorder-averaged quasienergy eigenstate correlator reproduces circular unitary ensemble structure to leading order in $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$1. That result implies a Floquet analogue of Berry’s conjecture for circuits, gives the CUE two-level correlator and ramp-plus-plateau spectral form factor, and yields ETH scaling for local operators, with $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$2 for few-body observables. Since $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$3 grows exponentially with system size, the off-diagonal matrix elements are exponentially small in system size, exactly as ETH requires (Liao et al., 2022).

Random-matrix classification in Floquet problems is subtler than in static ones because the relevant symmetry class is determined by $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$4, not by the instantaneous Hamiltonian at each moment of the drive. A periodically driven many-body system can therefore be in a different random-matrix class from the instantaneous Hamiltonian. A Floquet problem can be CUE even when $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$5 is always in the orthogonal class, COE even when $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$6 is always in the unitary class, thermalizing even when each instantaneous Hamiltonian is integrable, and effectively integrable at special commensurate frequencies although it thermalizes generically. The correct universal descriptor is the random-matrix class of the Floquet operator itself (Regnault et al., 2015).

An instructive complementary setting is the periodically driven fully connected Lipkin-Ising ferromagnet. There, Floquet thermalization to $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$7, Wigner-Dyson quasienergy statistics, ETH in the Floquet basis, and Hilbert-space delocalization all appear when the corresponding $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$8 classical dynamics is ergodic. When the classical dynamics is regular, observables still relax to a Floquet-diagonal steady value, but that value remains initial-state dependent and nonthermal (Russomanno et al., 2014).

3. Prethermalization, delayed heating, and quasiconserved observables

The most common deviation from immediate infinite-temperature heating is Floquet prethermalization. In high-frequency regimes, the drive generates an effective local Hamiltonian that acts as a quasiconserved quantity for exponentially long times. The system first relaxes with respect to this emergent Hamiltonian and only later heats further. In periodically kicked spin chains, this structure persists not only near the strict high-frequency limit but also near isolated finite-frequency commensurate points where part of the drive becomes exactly trivial. Around such points, the system exhibits a long-lived prethermal plateau, subsystem thermalization with respect to an effective Hamiltonian extracted from the inverse-frequency expansion, and eventual heating toward infinite temperature through processes not captured by the expansion itself (Fleckenstein et al., 2021).

A useful operator-space formulation of this phenomenon is based on infinite-temperature autocorrelations. By diagonalizing the infinite-time correlation matrix of local observables, one can isolate linearly independent quasiconserved directions. In two Floquet spin models, this procedure identifies a standard high-frequency energy-like quasiconserved observable and, in one case, an additional quasiconserved operator induced by a large global rotation. The latter persists at moderate frequency even when the conventional prethermal Hamiltonian is already less accurate, showing that delayed Floquet thermalization need not be organized by energy quasiconservation alone (Yin et al., 2020).

Recent flow-renormalization work near dynamical freezing gives a more detailed late-time picture. In a translationally invariant driven nonintegrable Ising chain, freezing is not an exact nonheating phase but an unstable prethermal fixed point with an additional approximate symmetry. Thermalization is delayed because the symmetry-violation measure is suppressed as $$U_F=\mathcal T\exp\!\left[-i\int_0^T H(t)\,dt\right]$$9 at freezing, but eventual heating occurs through nonperturbative instanton events in the flow. These instantons reorganize the many-body quasienergy structure through band-folding events and connect a sequence of unstable intermediate fixed points on the way to the thermal fixed point (Mukherjee et al., 2024).

A different proposal interprets prethermalization through the topology of a Krylov chain built in operator space. In that picture, a quasi-edge mode localized near the boundary of the Krylov chain plays the role of an approximate conserved quantity, and prethermalization is the tunneling of that quasi-edge mode through a local gap into the bulk. The same construction is used to define an effective prethermal Hamiltonian (Qi et al., 2024).

4. Diagnostics and operational characterizations

Floquet thermalization is diagnosed through a combination of spectral, eigenstate, dynamical, and reduced-state observables. The most common spectral quantity is the adjacent-gap ratio of quasienergies. In thermalizing Floquet systems, the relevant statistics are those of circular ensembles—typically COE or CUE depending on antiunitary symmetry—whereas Poisson statistics signal integrability or localization (Regnault et al., 2015, Zhang et al., 2016).

Eigenstate diagnostics include entanglement entropy, inverse participation ratios, and distributions of diagonal matrix elements. In the thermal phase of a Floquet model with no local conservation laws, eigenstate entanglement approaches the Page value for random pure states, the entanglement variance tends to zero, and imbalance observables quickly decay toward zero, reflecting efficient heating to infinite temperature (Zhang et al., 2016). In driven quasiperiodic chains and related nonergodic extended regimes, by contrast, Floquet eigenstates can be multifractal rather than fully ergodic, with intermediate level statistics and nontrivial fractal dimensions (Paul et al., 7 May 2026).

Dynamical probes include operator autocorrelations, subsystem entanglement growth, spectral form factors, and transport observables. In local random circuits, the quasienergy eigenstate correlator directly determines the spectral form factor and the thermalizing time dependence of observables, linking CUE eigenstate statistics to the universal late-time ramp-plus-plateau form of $2\pi/T$0 (Liao et al., 2022). In quasiperiodic Floquet Ising chains, autocorrelations distinguish ergodic, MBL, and many-body critical behavior, while the half-chain entanglement entropy separates logarithmic MBL growth from subdiffusive many-body critical growth and faster ergodic saturation (Paul et al., 7 May 2026).

A recent methodological extension combines reduced density matrices and work statistics. In a nonintegrable driven Ising chain, the quantum relative entropy $2\pi/T$1 for small subsystems tracks local thermalization, while coherence-sensitive characteristic functions of work and fluctuation-theorem diagnostics reproduce the same crossover structure. Low-frequency driving yields near-infinite-temperature behavior; intermediate frequency produces a coherence-rich prethermal crossover; and high-frequency driving is well described by a finite-$2\pi/T$2 effective thermal state over accessible times (Lin et al., 1 Jul 2026).

Experimental diagnostics can also be unusually local. In a Floquet-engineered dipolar NV ensemble, a disorder-enabled “disorder-order” protocol measures infinite-temperature local autocorrelators $2\pi/T$3, revealing local thermalization even where global polarization is symmetry-protected. This makes explicit that global memory retention and local equilibration need not coincide (Martin et al., 2022).

5. Mechanisms that suppress, delay, or reshape thermalization

Although infinite-temperature heating is generic, several mechanisms produce anomalous or incomplete Floquet thermalization. One class consists of localization-based phenomena. In disordered Floquet spin chains, the absence of conservation laws makes the thermal phase particularly clean, while the localized phase retains memory and Poisson level statistics (Zhang et al., 2016). In quasiperiodic driven Ising chains, lowering the frequency can suppress the MBL phase but replace it not by a fully ergodic state, but by a broad many-body critical or nonergodic extended regime with intermediate level statistics, slow autocorrelation decay, and subdiffusive entanglement growth (Paul et al., 7 May 2026).

A second class arises from strong driving and emergent constraints. In a clean interacting kicked Ising chain, thermalization sets in only below a threshold drive amplitude. Above that threshold, an emergent approximately conserved longitudinal magnetization underpins a frozen regime in which remanent magnetization remains finite in the infinite-time diagonal ensemble. The nonergodic sector is thermodynamically subdominant but has finite entropy, so this is neither standard Floquet-MBL nor ordinary high-frequency prethermalization (Haldar et al., 2018).

A third class is geometric or bottleneck-induced. Two fermionic chains coupled only by a periodically driven quantum point contact exhibit a critical driving frequency above which transport through the contact halts, the initial left-right density imbalance persists indefinitely, and Floquet ETH fails even with interactions and no disorder. The failure is encoded directly in the Floquet eigenstate structure, since diagonal matrix elements of the right-chain occupation remain broadly distributed rather than concentrating near the homogeneous infinite-temperature value (Dudinets et al., 2024).

A fourth class concerns symmetry-conserving circuit architectures. In $2\pi/T$4-conserving Floquet random circuits, the minimal nearest-neighbor spin-$2\pi/T$5 brickwork model with a single repeated layer is not robustly thermalizing at accessible sizes: it shows slow subdiffusive charge dynamics and parametrically slow entanglement saturation. Robust thermalization is restored by modest extensions such as longer-range gates, an enlarged on-site Hilbert space, or a larger Floquet period (Jonay et al., 2022).

Conversely, periodic driving can also enhance thermalization in otherwise constrained clean systems. In a one-dimensional Rydberg array with square-wave modulation, thermalization is strongest near reciprocal conditions $2\pi/T$6, where quasienergy statistics become COE-like, entanglement approaches the Page value, edge-mode correlations decay, and stroboscopic populations relax toward their infinite-temperature values (He et al., 15 May 2025).

6. Experimental platforms, rigorous extensions, and broader formulations

Experiments have directly resolved Floquet thermalization over wide timescales. In a bulk hyperpolarized solid of dipolar-coupled $2\pi/T$7 nuclei in diamond, pulsed spin-lock driving generates a long-lived prethermal plateau with a $2\pi/T$8 lifetime $2\pi/T$9, more than $H(t+T)=H(t)$0-fold longer than the free induction decay, while continuous readout reveals four regimes: an initial constrained transient, a prethermal plateau, unconstrained thermalization, and drift toward an infinite-temperature-like state. The experiment makes visible the crossover from emergent quasiconservation to eventual Floquet heating in a solid-state many-body system (Beatrez et al., 2021).

Floquet engineering has also become a tool for studying thermalization under engineered effective Hamiltonians rather than drive-induced heating per se. In a disordered three-dimensional NV-center ensemble, periodic pulse sequences realize tunable XXZ Hamiltonians, and local infinite-temperature autocorrelators reveal how anisotropy changes the shape and timescale of local relaxation. Near Ising, static local fields yield decay with stretching exponent $H(t+T)=H(t)$1; when flip-flops become important, the exponent is reduced toward the Markovian value $H(t+T)=H(t)$2 for $H(t+T)=H(t)$3 (Martin et al., 2022).

On the theoretical side, several recent works broaden the scope of Floquet thermalization beyond generic chaotic quantum lattices. In translationally invariant Clifford quantum cellular automata, sufficiently nonperiodic Floquet dynamics on an infinite lattice causes a broad class of initial states—including short-range entangled states close to equilibrium—to relax locally to the infinite-temperature state. A key distinction there is between strong thermalization, meaning convergence at all late times, and weak thermalization, meaning convergence for a density-1 subset of times (Kapustin et al., 1 Jan 2026). An analogous rigorous result has been obtained for a class of classical many-body Floquet systems of algebraic origin: if the Floquet map has the frequency-blowup property, then every URL initial state, including Gibbs states of sufficiently uniform local differentiable Hamiltonians, converges weakly to the Haar state, and the only obstruction is the existence of local observables periodic in time up to translation (Kapustin, 13 Mar 2026).

Taken together, these developments define Floquet thermalization as a hierarchy rather than a single phenomenon. At one extreme lie CUE- or COE-governed chaotic phases whose eigenstates satisfy Floquet ETH and heat to infinite temperature. At the other lie regimes stabilized by exact or approximate conservation laws, localization, bottlenecks, or topological and operator-space structures. Between them lie prethermal regimes in which effective Hamiltonians, quasi-edge modes, or unstable fixed points organize long-lived but ultimately transient nonthermal behavior. The central unifying theme is that the long-time fate of a driven many-body system is encoded simultaneously in quasienergy statistics, Floquet eigenstate structure, and the dynamical pathways by which local memory is lost—or retained.