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arXiv:2510.04137 · Annales Henri Poincaré

Polynomial Prethermal Lifetimes in Non Smoothly Driven Quantum Systems

Matteo Gallone, Beatrice Langella

Published Aug 10, 2026• 0 citations
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Abstract

Abstract We study the dynamics of a quantum many-body lattice system with a local Hamiltonian subjected to a quasi-periodic driving with finite regularity. For sufficiently large driving frequencies, we prove that the system remains in a prethermal state for times growing polynomially with the frequency, and we show the optimality of this bound by constructing an explicit example that nearly saturates it. Within this prethermal regime, the dynamics is captured by an effective time-independent local Hamiltonian close to the undriven one. The proof relies on a non convergent normal form scheme, combined with original smoothing techniques for finitely differentiable local operators, and Lieb–Robinson bounds.

DOI
10.1007/s00023-026-01714-y

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Abstract We study the dynamics of a quantum many-body lattice system with a local Hamiltonian subjected to a quasi-periodic driving with finite regularity. For sufficiently large driving frequencies, we prove that the system remains in a prethermal state for times growing polynomially with the frequency, and we show the optimality of this bound by constructing an explicit example that nearly saturates it. Within this prethermal regime, the dynamics is captured by an effective time-independent local Hamiltonian close to the undriven one. The proof relies on a non convergent normal form scheme, combined with original smoothing techniques for finitely differentiable local operators, and Lieb–Robinson bounds.
We investigate the periodically driven dynamics of many-body systems, either classical or quantum, finite-dimensional or mean-field, displaying an unbounded phase-space. Using the lattice ϕ4ϕ^4ϕ4 model and the ppp-spin spherical model as representative examples, we find that the inclusion of a smooth periodic drive atop an otherwise ergodic dynamics leads to a long-lived prethermalization, even at moderate driving frequencies. In specific asymptotic limits, we compute the corresponding prethermal Hamiltonian from an analytical perturbation scheme.
We establish some general dynamical properties of quantum many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasiconserved extensive quantity H∗{H}_{*}H∗​, which plays the role of an effective static Hamiltonian. The dynamics of the system (e.g., evolution of any local observable) is well approximated by the evolution with the Hamiltonian H∗{H}_{*}H∗​ up to time ${\ensuremath{\tau}}_{*}$, which is exponentially large in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where H∗{H}_{*}H∗​ is ergodic, the driven system prethermalizes to a thermal state described by H∗{H}_{*}H∗​ at intermediate times $t\ensuremath{\lesssim}{\ensuremath{\tau}}_{*}$, eventually heating up to an infinite-temperature state after times $t\ensuremath{\sim}{\ensuremath{\tau}}_{*}$. Our results indicate that rapidly driven many-body systems generically exhibit prethermalization and very slow heating. We briefly discuss implications for experiments which realize topological states by periodic driving.
Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency ν. We prove that up to a quasi-exponential time τ∗∼ecνlog3ν, the system barely absorbs energy. Instead, there is an effective local Hamiltonian Dˆ that governs the time evolution up to τ∗, and hence this effective Hamiltonian is a conserved quantity up to τ∗. Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi–Hubbard model where the interaction U is much larger than the hopping J. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time τ∗ that is (almost) exponential in U/J.
Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency ν\nuν. We prove that up to a quasi-exponential time τ∗∼ecνlog⁡3ν\tau_* \sim e^{c \frac{\nu}{\log^3 \nu}}τ∗​∼eclog3νν​, the system barely absorbs energy. Instead, there is an effective local Hamiltonian D^\hat DD^ that governs the time evolution up to τ∗\tau_*τ∗​, and hence this effective Hamiltonian is a conserved quantity up to τ∗\tau_*τ∗​. Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi-Hubbard model where the interaction UUU is much larger than the hopping JJJ. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time τ∗\tau_*τ∗​ that is (almost) exponential in U/JU/JU/J.
It is well understood that many-body systems driven at high frequency heat up only exponentially slowly and exhibit a long prethermalization regime. We prove rigorously that a certain relevant class of systems heat very slowly under weak periodic driving at intermediate frequency as well. This class of systems are those whose time-dependent, possibly translation-invariant, Hamiltonian is a weak perturbation of a sum of mutually commuting terms. This condition covers several periodically kicked systems that have been considered in the literature recently, in particular kicked Ising models. In contrast to the high-frequency regime, the prethermalization dynamics of our systems is in general not related to any time-independent effective Floquet Hamiltonian. Our results also have non-trivial implications for closed (time-independent) systems. We use the example of an Ising model with transversal and longitudinal field to show how they imply confinement of excitations. More generally, they show how "glassy" kinetically constrained models emerge naturally from simple many-body Hamiltonians, thus connecting to the topic of 'translation-invariant localization'.