Floquet Thermalization Conditions

• Floquet thermalization conditions are criteria governing how driven many-body systems relax to thermal ensembles, with drive frequency, conservation laws, and resonance effects playing key roles.
• Diagnostics such as spectral statistics, entanglement spectra, and operator dynamics help differentiate between infinite-temperature heating, prethermal regimes, and localization effects.
• These insights guide experimental designs and theoretical studies in quantum simulations, impacting our understanding of ergodicity, localization, and many-body dynamics in driven systems.

Floquet thermalization conditions are the conditions under which a periodically driven many-body system relaxes, in stroboscopic observables or local reduced states, to a thermal ensemble generated by the drive. In the literature, that endpoint is not unique: in closed ergodic systems without relevant conservation laws it is typically an infinite-temperature state; with conserved quantities it is a constrained infinite-temperature state; in high-frequency or open-system settings it can be a finite-temperature-like prethermal or Floquet-Gibbs state; and in several classes of models thermalization fails altogether because of localization, symmetry protection, integrability, or exact invariant subspaces. No universal theorem covering all quantum Floquet systems is presently available; instead, the subject is organized by model-dependent mechanisms, together with a few rigorous criteria in restricted classical and Clifford settings (Bera et al., 2024, Kapustin, 13 Mar 2026, Kapustin et al., 1 Jan 2026).

1. Floquet thermalization as driven equilibration

For a $T$-periodic Hamiltonian $H(t+T)=H(t)$, the one-period evolution operator is

$$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$

with quasienergies defined modulo $2\pi/T$. In finite systems with nondegenerate quasienergies, long-time stroboscopic observables are controlled by the Floquet diagonal ensemble, and the Floquet version of ETH states that diagonal matrix elements of local observables should become thermal within the relevant symmetry sector (Dudinets et al., 2024, Liu, 2014).

In generic interacting closed Floquet systems without extensive conservation laws, the expected endpoint is heating to a featureless infinite-temperature state. This expectation is explicit in the disordered Floquet Ising chain without conservation laws, where local inverse temperatures $\beta_j(n)$ are observed to decay toward zero throughout the numerically accessible ergodic regime, and in the periodically driven fully connected Ising ferromagnet, where ergodic dynamics yields $T=\infty$ ETH for intensive observables (Bera et al., 2024, Russomanno et al., 2014).

Conservation laws modify the endpoint rather than eliminating thermalization. In the driven fermionic transport problem with conserved particle number, the Floquet-ETH prediction is not a featureless state but a uniform-density state constrained by total $N$; for the right chain this gives

$\overline N_R=N\frac{L}{2L+1}.$

Thermalization in that setting therefore means homogenization subject to particle-number conservation, not unconstrained heating (Dudinets et al., 2024).

A different finite-system viewpoint appears when an effective static Hamiltonian $H_{\mathrm{eff}}$ can be constructed. In that case, period-averaged observables may be governed by ETH for $H_{\mathrm{eff}}$, so late-time values depend only on the initial energy with respect to $H(t+T)=H(t)$0, rather than on quasienergies modulo the drive frequency. This is a distinct notion from thermodynamic-limit infinite-temperature heating (Liu, 2014).

2. Diagnostics and operational criteria

The most common operational criterion is Floquet spectral statistics. For ordered quasienergies $H(t+T)=H(t)$1, the adjacent-gap ratio

$H(t+T)=H(t)$2

distinguishes random-matrix-like thermalization from integrable or localized behavior. Benchmark values used repeatedly are $H(t+T)=H(t)$3 for Poisson, $H(t+T)=H(t)$4 for GOE/COE-type statistics, and $H(t+T)=H(t)$5 for GUE/CUE-type statistics (Geraedts et al., 2016, Regnault et al., 2015).

Floquet thermalization is also diagnosed in the operator and eigenstate sectors. In disordered interacting Floquet systems, diagonal and off-diagonal matrix elements of local operators can violate standard ETH while still shrinking with system size. In the subdiffusive regime of a disordered driven Heisenberg chain, the paper reports

$H(t+T)=H(t)$6

linking modified ETH directly to the dynamical decay exponent $H(t+T)=H(t)$7 (Roy et al., 2018).

Entanglement-based diagnostics are particularly informative in Floquet systems because infinite-temperature ETH makes the leading reduced density matrix trivial. The entanglement spectrum can therefore probe structure beyond standard ETH: in thermal Floquet phases it shows random-matrix level statistics, while in Floquet-MBL phases it is semi-Poisson. The random-matrix class of the Floquet entanglement spectrum can depend on drive symmetries and even on the chosen origin of time (Geraedts et al., 2016).

Local equilibration can be tested directly from reduced density matrices. In a periodically driven nonintegrable Ising chain, subsystem thermalization is quantified by the relative entropy

$H(t+T)=H(t)$8

while work statistics provide a parallel diagnostic through TPM and coherence-preserving characteristic functions. The same frequency-controlled crossover appears in both: low-frequency efficient heating, an intermediate prethermal regime with persistent coherence, and a high-frequency finite-temperature-like regime (Lin et al., 1 Jul 2026).

Other diagnostics are explicitly model dependent. They include bond-resolved effective inverse temperatures $H(t+T)=H(t)$9 and local thermalization times $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$0 in a disordered Floquet Ising chain (Bera et al., 2024); current stoppage and persistent imbalance in a driven quantum point contact (Dudinets et al., 2024); inverse participation ratios and effective dimensions in Floquet eigenbases (Jonay et al., 2022, C et al., 26 Nov 2025); and entanglement growth laws, from logarithmic growth in localized regimes to Page-like saturation in ergodic ones (Nagao et al., 13 Mar 2026, Paul et al., 7 May 2026).

3. Frequency, amplitude, and commensurability

Frequency is the most widely used control parameter, but the literature shows that its role is not uniform across models. In a periodically driven nonintegrable Ising chain, low frequency yields efficient heating toward infinite temperature, intermediate frequency supports a prethermal crossover regime, and high frequency produces a finite-temperature-like late-time state governed by an effective or averaged Hamiltonian (Lin et al., 1 Jul 2026). In mesoscopic chaotic Floquet systems, the same theme is refined into scaling regimes: the ladder ensemble becomes resolvable at $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$1, generic product states resolve it only when $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$2, and the breakdown of Floquet thermalization itself occurs only at $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$3 (Morningstar et al., 2022).

A particularly sharp threshold appears in the driven fermionic quantum point contact. There, Floquet thermalization holds only below a critical frequency approximately equal to the single-particle bandwidth scale, $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$4 for the interacting cases studied with $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$5. For $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$6, the current through the contact is nonzero and the density profile homogenizes; for $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$7, the contact becomes effectively insulating, $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$8, and diagonal Floquet matrix elements of $$U(T)=\mathcal T\exp\!\left(-i\int_0^T H(t)\,dt\right)=e^{-iH_FT},$$9 spread across nearly the full allowed range instead of concentrating near the uniform value (Dudinets et al., 2024).

Commensurability can either induce or destroy thermalization. In a clean one-dimensional Rydberg array with square-wave driving, the authors identify “reciprocal Floquet thermalization” when

$2\pi/T$0

or exactly $2\pi/T$1 with $2\pi/T$2. Near these windows, level statistics approach COE, $2\pi/T$3 relaxes to $2\pi/T$4, and entanglement reaches a Page-like value; away from them, the dynamics is much more integrable or localized (He et al., 15 May 2025). By contrast, in a two-bang disordered spin chain, special commensurate periods

$2\pi/T$5

make the Floquet operator simplify to decoupled on-site rotations, restoring integrable behavior inside an otherwise thermalizing Floquet protocol (Regnault et al., 2015).

Amplitude can dominate over frequency. In the disorder-free driven Ising chain of “Onset of Floquet Thermalisation,” weak drive amplitude yields heating to the infinite-temperature ensemble, whereas above a threshold amplitude $2\pi/T$6 the system develops nonzero remanent $2\pi/T$7-magnetization up to infinite time. The threshold scales approximately with the transverse field, $2\pi/T$8, so the control parameter is the drive amplitude relative to the noncommuting static terms rather than a high-frequency limit (Haldar et al., 2018).

4. Disorder, transport, and spatially heterogeneous heating

Disorder does not impose a single Floquet outcome. In a disordered periodically driven Ising chain without conservation laws, thermalization persists throughout the numerically accessible ergodic regime, but it is strongly inhomogeneous: different bonds exhibit widely different local inverse-temperature decay times, the disorder-averaged decay is stretched exponential, and the disorder-averaged diagonal entropy acts as an internal clock that collapses local relaxation curves across disorder strengths (Bera et al., 2024).

When a conserved quantity is present, disorder can produce anomalous thermalization rather than simple failure of ETH. In the driven disordered Heisenberg chain with conserved $2\pi/T$9, the ergodic phase exhibits subdiffusive transport and non-Gaussian distributions of Floquet-basis matrix elements. The system still approaches the periodic infinite-temperature ensemble, but only a modified ETH survives, with the scaling relation

$\beta_j(n)$0

between the finite-size enhancement of off-diagonal matrix elements and the autocorrelation decay exponent (Roy et al., 2018).

In larger quasiperiodic Floquet Ising systems implemented on quantum hardware, the distinction is operationally between an ergodic–MBL crossover rather than a proven asymptotic transition. Weak quasiperiodic strength $\beta_j(n)$1 gives rapid decay of autocorrelation and QFI approaching the Haar-random benchmark $\beta_j(n)$2, while strong $\beta_j(n)$3 yields persistent autocorrelation over thousands of cycles and logarithmic QFI growth, consistent with MBL-like suppression of Floquet heating. The reported crossover scales are $\beta_j(n)$4 in one dimension and $\beta_j(n)$5 in two dimensions (Nagao et al., 13 Mar 2026).

Quasiperiodic driving can also suppress thermalization without producing Floquet-MBL. In the kicked quasiperiodic Ising chain with binary drive, high frequency supports coexisting ergodic, MBL, and many-body critical phases, but at moderate frequency the MBL phase is destroyed and replaced by a broad MBC regime. The crucial condition is $\beta_j(n)$6, where near-zero transverse fields $\beta_j(n)$7 fragment Fock-space connectivity: for $\beta_j(n)$8 the system is ergodic, while for $\beta_j(n)$9 it becomes nonergodic extended rather than thermal (Paul et al., 7 May 2026).

Even without disorder, a conserved charge and minimal circuit architecture can delay thermalization severely. In $T=\infty$0-conserving Floquet random circuits, the minimal $T=\infty$1, $T=\infty$2, $T=\infty$3 brickwork model is not robustly thermalizing at accessible sizes and shows long-lived subdiffusive dynamics. Robust thermalization is restored by any of three modifications: three-site gates, an enlarged local Hilbert space $T=\infty$4, or a longer Floquet period $T=\infty$5 with two independent gate layers (Jonay et al., 2022).

5. Symmetry, topology, and exact obstructions

In some classes of models the relevant condition is not disorder or frequency alone but the structure of invariant subspaces or the dynamics of collective variables. In the periodically driven fully connected Lipkin-Ising ferromagnet, Floquet thermalization occurs precisely when the $T=\infty$6 classical dynamics is ergodic or chaotic; in that regime quasienergies show Wigner-Dyson statistics, Floquet states are delocalized in Hilbert space, and observables relax to the $T=\infty$7 value independently of the initial state. When the classical dynamics is regular, the system relaxes only to an initial-state-dependent Floquet diagonal ensemble (Russomanno et al., 2014).

Exact invariant subspaces can produce measure-zero violations of Floquet ETH inside otherwise thermalizing models. In the driven Rydberg-blockaded PXP construction, a four-dimensional subspace $T=\infty$8 is exactly preserved by the Floquet operator for arbitrary nonzero pulse durations in the main protocol. States in $T=\infty$9 form exact Floquet scars with low entanglement and nonthermal observables, while generic states outside $N$0, including special states familiar from the static PXP problem, heat to infinite temperature (Mizuta et al., 2020).

A more recent proposal assigns Floquet thermalization conditions to topology in Krylov space. There, the Floquet Liouvillian generated from the period-averaged Hamiltonian defines a chiral Krylov chain. A nontrivial Krylov-chain topology with a localized zero edge mode implies a finite-temperature state governed by an unfolded effective Hamiltonian, whereas a trivial or gapless chain implies heating to infinite temperature. In the thermodynamic limit, the paper argues that high-frequency prethermalization can be viewed as tunnelling of a quasi-edge mode through a local Krylov-space gap (Qi et al., 2024).

Rigorous criteria exist in Clifford and classical algebraic settings. For translationally invariant Clifford-Floquet dynamics, weak diffusivity is equivalent to the absence of nonzero solutions of

$N$1

for all $N$2 and $N$3; equivalently, there are no translation-periodic Pauli observables or “solitons.” Under that condition, P-generic product states are weakly thermalized to infinite temperature, and if the QCA is strongly diffusive the thermalization is strong. The same mechanism extends, with an explicit smallness condition, to short-range-entangled states sufficiently close to equilibrium (Kapustin et al., 1 Jan 2026).

An exact classical analogue exists for a large family of local algebraic torus automata. There, thermalization to Haar measure for all URL initial states is equivalent to “frequency blowup,” and frequency blowup is in turn equivalent to the absence of nonconstant local or quasilocal observables satisfying

$N$4

or, algebraically,

$N$5

Within that class, periodic local observables up to translation are the only obstruction to thermalization (Kapustin, 13 Mar 2026).

Long-range Floquet models interpolate between symmetry-restricted and thermalizing regimes. In the kicked power-law Ising chain, $N$6 preserves permutation symmetry and confines dynamics to the permutation-symmetric subspace; intermediate $N$7 breaks that symmetry strongly enough to produce COE statistics, Page-like entanglement, and $N$8 values matching random states in the full Hilbert space; and large $N$9 approaches the integrable kicked Ising limit. Larger driving period $\overline N_R=N\frac{L}{2L+1}.$0 shifts the onset of thermalization to smaller $\overline N_R=N\frac{L}{2L+1}.$1 and extends the thermalizing window (C et al., 26 Nov 2025).

6. Open systems, mesoscopic refinements, and the scope of universality

Open periodically driven systems obey a different set of conditions. In the weak-coupling Floquet master-equation framework, a Floquet-Gibbs steady state at the bath temperature is obtained under three sufficient conditions: the static system Hamiltonian is bounded and $\overline N_R=N\frac{L}{2L+1}.$2; the driving Hamiltonians commute at different times,

$\overline N_R=N\frac{L}{2L+1}.$3

and the drive commutes with the system-bath interaction,

$\overline N_R=N\frac{L}{2L+1}.$4

Under these restrictions, sideband-assisted bath processes are suppressed and detailed balance holds in the Floquet basis (Shirai et al., 2014).

For finite isolated systems, an effective-Hamiltonian description can also organize Floquet thermalization. If a periodic unitary transformation exists that renders the problem time independent, and if the resulting $\overline N_R=N\frac{L}{2L+1}.$5 is nonintegrable and satisfies ETH, then long-time period-averaged observables are described by a microcanonical ensemble of $\overline N_R=N\frac{L}{2L+1}.$6 at the initial effective energy. This is a finite-system mechanism and should be distinguished from thermodynamic-limit heating to infinite temperature (Liu, 2014).

Representative conditions reported in the literature can be summarized compactly:

Setting	Reported condition	Reported endpoint
Generic closed ergodic Floquet system without conservation laws	Ergodic regime under periodic drive	Infinite-temperature state
Driven fermionic QPC	$\overline N_R=N\frac{L}{2L+1}.$7 / $\overline N_R=N\frac{L}{2L+1}.$8	Homogeneous constrained density / persistent imbalance
Clean Rydberg array	$\overline N_R=N\frac{L}{2L+1}.$9	Chaos and rapid thermalization
Mesoscopic chaotic Floquet system	$H_{\mathrm{eff}}$0, $H_{\mathrm{eff}}$1, $H_{\mathrm{eff}}$2	Ladder ensemble onset, state-resolution scale, breakdown scale
Open weak-coupling Floquet system	Conditions (i)–(iii) above	Floquet-Gibbs state

These results suggest that “Floquet thermalization conditions” are best understood not as a single universal threshold but as a hierarchy of constraints involving drive frequency and amplitude, resonance structure, conserved quantities, disorder or quasiperiodicity, operator spreading, symmetry-breaking scales, and the presence or absence of periodic local obstructions. Only in special settings—such as algebraic classical Floquet systems or Clifford QCAs—does the literature presently provide an if-and-only-if characterization (Morningstar et al., 2022, Kapustin, 13 Mar 2026, Kapustin et al., 1 Jan 2026).