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arXiv:2603.13631

On thermalization in many-body classical Floquet systems

Anton Kapustin

Published Mar 13, 2026• 0 citations
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Abstract

It is expected that a generic closed many-body system prepared in a well-behaved initial state and subjected to a periodic drive will eventually thermalize, i.e. approach the state of maximal entropy. This property, while compatible with and even demanded by the physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. We describe an infinite set of classical many-body Floquet systems of algebraic origin for which thermalization of very general initial states can be proved. For example, we show that a Gibbs state of any sufficiently uniform local differentiable Hamiltonian heats up to infinite temperature at long times. We show that in agreement with the physical intuition, the only obstruction to thermalization is the existence of local observables which are periodic in time.

DOI
10.48550/arxiv.2603.13631

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It is expected that a generic closed many-body system prepared in a well-behaved initial state and subjected to a periodic drive will eventually thermalize, i.e. approach the state of maximal entropy. This property, while compatible with and even demanded by the physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. We describe an infinite set of classical many-body Floquet systems of algebraic origin for which thermalization of very general initial states can be proved. For example, we show that a Gibbs state of any sufficiently uniform local differentiable Hamiltonian heats up to infinite temperature at long times. We show that in agreement with the physical intuition, the only obstruction to thermalization is the existence of local observables which are periodic in time.
Abstract A Floquet quantum system is governed by a Hamiltonian that is periodic in time. Consider the space of piecewise time-independent Floquet systems with (geometrically) local interactions. We prove that for all but a measure zero set of systems in this space, starting from a random product state, many properties (including expectation values of observables and the entanglement entropy of a macroscopically large subsystem) at long times are approximately periodic with the same period as the Hamiltonian. Thus, in almost every Floquet system of arbitrarily large but finite size, discrete time-crystalline behavior does not persist to strictly infinite time.
Periodically driven many-body systems generally heat towards a featureless 'infinite-temperature' state. As an alternative to uniform heating in a clean system, here we establish a Floquet superheating regime, where fast heating nucleates at ''hot spots" generated by rare fluctuations in the local energy with respect to an appropriate effective Hamiltonian. Striking macroscopic consequences include exceptionally long-lived prethermalization and non-ergodic bimodal distributions of macroscopic observables. Superheating is predicated on a heating rate depending strongly on the local fluctuation; in our example, this is supplied by a sharp state-selective spin-echo, where the energy absorption is strongly suppressed for low-energy states, while thermal fluctuations open up excessive heating channels. A simple phenomenological theory is developed to show the existence of a critical droplet size, which incorporates heating by the driving field as well as the heat current out of the droplet. Our results shine light on a new heating mechanism and suggest new routes towards stabilizing non-equilibrium phases of matter in driven systems.
Interacting many-body quantum systems are believed to heat up unboundedly under a periodic drive as there is no local conservation law to prevent this from happening. Here, the authors show that this belief may not be generally true. Their results indicate that under strong periodic driving quantum interference may suppress the heating and may lead to a nonthermal state, remembering properties of the initial state until infinite time. This is accounted for by the emergence of an approximately conserved local quantity. The freezing is more pronounced for smaller drive frequencies, putting it beyond the scope of conventionally used high-frequency expansions.
Exploiting the rich phenomenology of periodically driven many-body systems is notoriously hindered by persistent heating in both the classical and the quantum realm. Here, we investigate to what extent coupling to a large thermal reservoir makes stabilization of a nontrivial steady state possible. To this end, we model both the system and the reservoir as classical spin chains where driving is applied through a rotating magnetic field, and we simulate the Hamiltonian dynamics of this setup. We find that the intuitive limits of infinite frequency and vanishing frequency, where the system dynamics is governed by the average and the instantaneous Hamiltonian, respectively, can be smoothly extended into entire regimes separated only by a small crossover region. At high frequencies, the driven system stroboscopically attains a Floquet-type Gibbs state at the reservoir temperature. At low frequencies, a global synchronized Gibbs state emerges, whose temperature may depart significantly from the initial temperature of the reservoir. Although our analysis in some parts relies on the specific properties of our setup, we argue that much of its phenomenology could be generic.
Floquet engineering is a powerful manipulation method in modern quantum technology. However, unwanted heating is the main challenge of Floquet engineering, therefore the Floquet thermalization has attracting considerable attentions recently. In this work, we investigate thermalization of periodically driven many-body systems through the lens of Krylov complexity, and find a topological origin of different thermalization behaviors. We demonstrate that If the topology of the Krylov chain is nontrivial, a periodically driven system will reach a state with finite temperature. When the Krylov chain is topologically trivial, the system will be heated to infinite temperature. We further show that the prethermalization can be understood as the tunnelling process of a quasi-edge mode through the local gap on Krylov chain. This picture provides a systematically method to obtain the effective prethermal Hamiltonian.
In isolated quantum many-body systems periodically driven in time, the asymptotic dynamics at late times can exhibit distinct behavior such as thermalization or dynamical freezing. Understanding the properties of and the convergence towards infinite-time (nonequilibrium) steady states, however, remains a challenging endeavor. We propose a physically motivated Krylov space perspective on Floquet thermalization which offers a natural framework to study rates of convergence towards steady states and, simultaneously, an efficient numerical algorithm to evaluate infinite-time averages of observables within the diagonal ensemble. The effectiveness of our algorithm is demonstrated by applying it to the periodically driven mixed-field Ising model, reaching system sizes of up to 30 spins. Our method successfully resolves the transition between the ergodic and dynamically frozen phases and provides insight into the nature of the Floquet eigenstates across the phase diagram. Furthermore, we show that the long-time behavior is encoded within the localization properties of the Ritz vectors under the Floquet evolution, providing an accurate diagnostic of ergodicity.