A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems

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 $\nu$ . We prove that up to a quasi-exponential time $\tau_* \sim e^{c \frac{\nu}{\log^3 \nu}}$ , the system barely absorbs energy. Instead, there is an effective local Hamiltonian $\hat D$ that governs the time evolution up to $\tau_*$ , and hence this effective Hamiltonian is a conserved quantity up to $\tau_*$ . 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 $\tau_*$ 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 $\nu$. We prove that up to a quasi-exponential time $\tau_* \sim e^{c \frac{\nu}{\log^3 \nu}}$, the system barely absorbs energy. Instead, there is an effective local Hamiltonian $\hat D$ that governs the time evolution up to $\tau_*$, and hence this effective Hamiltonian is a conserved quantity up to $\tau_*$. 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 $\tau_*$ that is (almost) exponential in $U/J$.

Sep 17, 20151 citation

Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems

Dmitry A. Abanin, Wojciech De Roeck, Wen Wei Ho +1 more

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 ${H}_{*}$ , 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 ${H}_{*}$ 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 ${H}_{*}$ is ergodic, the driven system prethermalizes to a thermal state described by ${H}_{*}$ 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.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 ${H}_{*}$, 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 ${H}_{*}$ 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 ${H}_{*}$ is ergodic, the driven system prethermalizes to a thermal state described by ${H}_{*}$ 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.

arXiv:1510.03405Physical review. B./Physical review. BJan 27, 2017362 citations

Prethermalization without Temperature

David J. Luitz, Roderich Moessner, Shivaji L. Sondhi +1 more

While a clean, driven system generically absorbs energy until it reaches "infinite temperature," it may do so very slowly exhibiting what is known as a prethermal regime. Here, we show that the emergence of an additional approximately conserved quantity in a periodically driven (Floquet) system can give rise to an analogous long-lived regime. This can allow for nontrivial dynamics, even from initial states that are at a high or infinite temperature with respect to an effective Hamiltonian governing the prethermal dynamics. We present concrete settings with such a prethernial regime, one with a period-doubled (time-crystalline) response. We also present a direct diagnostic to distinguish this prethermal phenomenon from its infinitely long-lived many-body localized cousin. We apply these insights to a model of the recent NMR experiments by Rovny et al. [Phys. Rev. Lett. 120, 180603 (2018)] which, intriguingly, detected signatures of a Floquet time crystal in a clean three-dimensional material. We show that a mild but subtle variation of their driving protocol can increase the lifetime of the time-crystalline signal by orders of magnitude.While a clean, driven system generically absorbs energy until it reaches "infinite temperature," it may do so very slowly exhibiting what is known as a prethermal regime. Here, we show that the emergence of an additional approximately conserved quantity in a periodically driven (Floquet) system can give rise to an analogous long-lived regime. This can allow for nontrivial dynamics, even from initial states that are at a high or infinite temperature with respect to an effective Hamiltonian governing the prethermal dynamics. We present concrete settings with such a prethernial regime, one with a period-doubled (time-crystalline) response. We also present a direct diagnostic to distinguish this prethermal phenomenon from its infinitely long-lived many-body localized cousin. We apply these insights to a model of the recent NMR experiments by Rovny et al. [Phys. Rev. Lett. 120, 180603 (2018)] which, intriguingly, detected signatures of a Floquet time crystal in a clean three-dimensional material. We show that a mild but subtle variation of their driving protocol can increase the lifetime of the time-crystalline signal by orders of magnitude.

Apr 1, 202030 citations

Fractionalized prethermalization in the one-dimensional Hubbard model

Anton Romen, Johannes Knolle, Michael Knap

Prethermalization phenomena in driven systems are generally understood via a local Floquet Hamiltonian obtained from a high-frequency expansion. Remarkably, recently it has been shown that a driven Kitaev spin liquid with fractionalized excitations can realize a quasistationary state that is not captured by this paradigm. Instead distinct types of fractionalized excitations are characterized by vastly different temperatures—a phenomenon dubbed . In our work, we analyze fractionalized prethermalization in a driven one-dimensional Hubbard model at strong coupling which hosts spin-charge fractionalization. At intermediate frequencies quasisteady states emerge which are characterized by a low spin and high charge temperature with lifetimes set by two competing processes: the lifetime of the quasiparticles determined by Fermi's Golden rule and the exponentially long lifetime of a Floquet prethermal plateau. We classify drives into three categories, each giving rise to distinct (fractional) prethermalization dynamics. Resorting to a time-dependent variant of the Schrieffer-Wolff transformation, we systematically analyze how these drive categories are linked to the underlying driven Hubbard model, thereby providing a general understanding of the emergent thermalization dynamics. We discuss routes toward an experimental realization of this phenomenon in quantum simulation platforms.Prethermalization phenomena in driven systems are generally understood via a local Floquet Hamiltonian obtained from a high-frequency expansion. Remarkably, recently it has been shown that a driven Kitaev spin liquid with fractionalized excitations can realize a quasistationary state that is not captured by this paradigm. Instead distinct types of fractionalized excitations are characterized by vastly different temperatures—a phenomenon dubbed . In our work, we analyze fractionalized prethermalization in a driven one-dimensional Hubbard model at strong coupling which hosts spin-charge fractionalization. At intermediate frequencies quasisteady states emerge which are characterized by a low spin and high charge temperature with lifetimes set by two competing processes: the lifetime of the quasiparticles determined by Fermi's Golden rule and the exponentially long lifetime of a Floquet prethermal plateau. We classify drives into three categories, each giving rise to distinct (fractional) prethermalization dynamics. Resorting to a time-dependent variant of the Schrieffer-Wolff transformation, we systematically analyze how these drive categories are linked to the underlying driven Hubbard model, thereby providing a general understanding of the emergent thermalization dynamics. We discuss routes toward an experimental realization of this phenomenon in quantum simulation platforms.

arXiv:2502.09708Physical review. B./Physical review. BNov 20, 20250 citations

Polynomial Prethermal Lifetimes in Non Smoothly Driven Quantum Systems

Matteo Gallone, Beatrice Langella

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.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.

arXiv:2510.04137Annales Henri PoincaréAug 10, 20260 citations

Floquet prethermalization in periodically driven classical spin systems

Takashi Mori

It is proved that the energy absorption in a periodically driven classical spin system is exponentially slow in frequency, which results in a two-step relaxation called the Floquet prethermalization. This result is shown by establishing the classical limit of the quantum spin dynamics. The Floquet prethermal state is well described by the Gibbs ensemble with respect to the static effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion. On the other hand, the same effective Hamiltonian does not reproduce the local dynamics for an exponentially long time. This is due to the chaoticity of classical dynamics and in stark contrast to quantum spin systems, in which the effective Hamiltonian well reproduces the exact quantum dynamics up to an exponentially long time.It is proved that the energy absorption in a periodically driven classical spin system is exponentially slow in frequency, which results in a two-step relaxation called the Floquet prethermalization. This result is shown by establishing the classical limit of the quantum spin dynamics. The Floquet prethermal state is well described by the Gibbs ensemble with respect to the static effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion. On the other hand, the same effective Hamiltonian does not reproduce the local dynamics for an exponentially long time. This is due to the chaoticity of classical dynamics and in stark contrast to quantum spin systems, in which the effective Hamiltonian well reproduces the exact quantum dynamics up to an exponentially long time.

arXiv:1804.02165Physical review. B./Physical review. BSep 10, 201861 citations

\nu

. We prove that up to a quasi-exponential time

\tau_* \sim e^{c \frac{\nu}{\log^3 \nu}}

, the system barely absorbs energy. Instead, there is an effective local Hamiltonian

\hat D

that governs the time evolution up to

\tau_*

, and hence this effective Hamiltonian is a conserved quantity up to

\tau_*

. 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

\tau_*

that is (almost) exponential in

U/J

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}_{*}

, 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}_{*}

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}_{*}

is ergodic, the driven system prethermalizes to a thermal state described by

{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.