Prethermal Phase in Quantum Systems

• Prethermal phase is a long-lived non-equilibrium state characterized by a rapid initial relaxation followed by a quasi-stationary plateau with persistent order.
• Studies show that emerging phenomena such as discrete time crystalline behavior and symmetry breaking arise from tailored interactions and precise drive conditions.
• The duration of prethermal phases is governed by weak perturbations, long-range interactions, and high driving frequencies, offering tunable pathways for non-equilibrium order.

A prethermal phase refers to a long-lived, non-equilibrium state that emerges in an isolated quantum or classical system following a rapid parameter change (quench) or under periodic driving, before the system ultimately relaxes to thermal equilibrium. In prethermal regimes, physical observables settle onto quasi-stationary plateaus that can exhibit properties—such as persistent order or symmetry breaking—not allowed in the true thermal state. Prethermalization is thus characterized by a dynamical separation of time scales: a fast relaxation to the prethermal state, followed by eventual slow drift toward thermal equilibrium. This phenomenon has been observed and rigorously characterized in a variety of integrable, near-integrable, and driven many-body systems.

1. Dynamical Formation and Separation of Time Scales

Following a quantum quench in the one-dimensional transverse-field Ising model with long-range interactions decaying as $1/|i-j|^\alpha$, prethermalization manifests as a rapid initial decay of the order parameter (such as the magnetization)

$$m(t) = \frac{1}{L} \sum_j \langle \psi_i(t) | \sigma_j^z | \psi_i(t) \rangle,$$

from its initial value, followed by a well-defined plateau at a prethermal value $\tilde{m}$ differing from both the initial and thermal values. This plateau persists over an extended intermediate time window before thermalization sets in (Halimeh et al., 2016). The fast approach to the prethermal regime is attributed to dephasing and dispersive spreading of low-density excitations, which is especially pronounced for small quench amplitudes $h_f - h_i$, where the system retains a memory of its symmetry-broken initial state for much longer than microscopic time scales.

This separation into two dynamical regimes—fast prethermalization, slow thermalization—is a universal property of weakly non-integrable or weakly driven systems, and arises generically when a dominant interaction or driving term is present (quantified by a small perturbation $\lambda$). In the case of integrable models perturbed by a weak generic $V$, a rigorous bound shows that the system's time evolution remains close to the integrable (and typically non-thermal) evolution for times $t \ll \hbar/\Gamma \sim \lambda^{-2}$, with a crossover to thermalization only at longer times, establishing the "typicality" of prethermalization for large classes of weakly perturbed many-body systems (Reimann et al., 2019).

2. Key Features and Order in the Prethermal Phase

The prethermal phase is distinguished by quasi-stationarity and the emergence or persistence of order parameters that would otherwise vanish in equilibrium. Simulations in the long-range Ising chain show that the plateau value $\tilde{m}$ of the magnetization remains nonzero over the prethermal window, even for $\alpha$ values where no finite-temperature ordered phase exists thermodynamically (Halimeh et al., 2016). The prethermal regime thus supports "persistent" symmetry breaking out of equilibrium: for instance, the system dynamically stabilizes a ferromagnetic phase, retaining nonzero magnetization well after the fast dephasing processes have subsided and prior to the entropic mixing of excitations that ultimately restores the symmetry required by thermal statistics.

In models with periodic driving (Floquet systems), the prethermal phase can exhibit time-crystalline order—manifested as a robust subharmonic (e.g., period-doubled) response in the magnetization—even without disorder or many-body localization (Machado et al., 2019, Kyprianidis et al., 2021, Zeng et al., 2017). In this context, prethermal phases are identified by long-lived, synchronized oscillations in order-parameter autocorrelations and by emergent dynamical plateaus in entanglement or mutual information, persisting exponentially long in the ratio of driving frequency to local energy scales.

These features are exposed not just in conventional observables but also in higher order correlations. The emergence of robust prethermal nematic phases (where orientational order survives even after stripe order melts) (Jin et al., 2022), as well as in unpolarized prethermal DTCs where period-doubled autocorrelations exist despite vanishing instantaneous order parameters (Yokota et al., 16 Jan 2025), reveals the role of quantum and classical fluctuations as stabilizing agents for nonequilibrium order.

3. Role of Interaction Range, Driving, and Perturbations

The structure and longevity of the prethermal phase is controlled by system-specific parameters:

• Long-range interactions ($1/r^\alpha$) in the Ising model enhance prethermal stability (Halimeh et al., 2016). While equilibrium order is only possible for $\alpha \leq 2$, prethermal symmetry-broken plateaus persist for larger $\alpha$ due to slow relaxation dynamics associated with the “mean-field-like” nature of the interactions. For $\alpha \gg 2$ (short-range limit), the prethermal plateau rapidly decays, indicating a crossover to trivial thermal dynamics.
• Periodically driven systems (Floquet systems) exhibit prethermal phases when the drive frequency $\omega$ far exceeds the local interaction scale, suppressing heating processes exponentially in $\omega/J_{\text{local}}$. Here, the system is governed by a prethermal Floquet Hamiltonian up to times $\tau^* \sim \exp(\omega/J_{\text{local}})$, within which emergent symmetries can stabilize non-equilibrium orders, including discrete time crystals (PDTC) free of disorder (Machado et al., 2019, Kyprianidis et al., 2021, Banerjee et al., 30 Jul 2024).
• Perturbations and weakly broken integrability: In the presence of weak integrability-breaking terms, prethermalization emerges as a nearly universal relaxation scenario. Mathematically, the time scale for the crossover to thermalization is set by the square of the perturbation strength $\Gamma \propto \lambda^2$ (Reimann et al., 2019), validating the two-stage relaxation picture.

4. Phase Transitions and Prethermal Criticality

Dynamical phase transitions in the prethermal regime occur as nonequilibrium analogs of equilibrium phase transitions, but with distinct critical points and scaling. In long-range Ising models, the prethermal-to-disordered transition (identified by a vanishing plateau $\tilde{m}$ as a function of the final field $h_f$) occurs at a "dynamical" critical point $\tilde{h}_c$ that is generally shifted from the equilibrium transition field $h_c(\alpha)$ and strongly dependent on both the range parameter $\alpha$ and the initial state (Halimeh et al., 2016). The critical behavior in this regime is determined by the interplay between the memory of the initial condition and the quench amplitude rather than by equilibrium thermodynamic considerations.

Self-similar scaling and universal dynamical crossovers in prethermal order-parameter dynamics have been analytically characterized near quantum critical points (Dağ et al., 2021). In this setting, the prethermal regime demonstrates scaling

$$\delta C_r(t, h_n) = C_r^{qs}(|h_n|) f_{\Delta, h_i}(h_n t)$$

where $h_n$ is the reduced distance to criticality, and the scaling function $f$ captures the universal interpolation between initial decay and quasi-stationary behavior. The duration of the prethermal plateau thus diverges as the critical point is approached and the gap closes, making dynamics in this regime a sensitive probe of underlying criticality.

5. Distinction from Thermal Equilibrium and Ultimate Fate

Prethermal phases are intrinsically nonequilibrium and are not described by conventional thermal ensembles except perhaps within restricted symmetry or conservation sectors. Observables in prethermal plateaus often take values forbidden in the thermal state: long-lived magnetization or nematic order in regimes where the equilibrium expectation is zero; or finite chiral currents and time-crystalline oscillations in circumstances where equilibrium predicts decay to zero (Paul et al., 15 Apr 2025, Jin et al., 2022).

Thermalization, when it occurs, is driven by processes not captured at the effective (prethermal) level—such as rare many-body resonances, higher-order Floquet corrections, or processes that violate emergent conservation laws. For example, in systems with confinement, the prethermal plateau corresponds to a sector with fixed “meson” number, while ultimate thermalization requires extremely rare processes analogous to the Schwinger effect that violate this conservation (Birnkammer et al., 2022).

Prethermal quasiconserved observables can be quantitatively tracked via infinite-temperature correlations and autocorrelators, where a slow decay signals the crossover from prethermal protection to full thermal mixing (Yin et al., 2020). In disordered systems, the prethermal many-body localized regime is marked by a hierarchy of stretched exponential decays with time scales (correlated to disorder strength) that grow exponentially, but that ultimately vanish in the thermodynamic limit (Long et al., 2022).

6. Mathematical Structure and Rigorous Frameworks

Quantitative understanding of prethermal phases relies on both analytical and numerical approaches:

• High-order Floquet-Magnus and Schrieffer–Wolff expansions yield effective prethermal Hamiltonians and emergent conservation laws, valid up to exponentially long times in the relevant drive or interaction scale (Machado et al., 2019, Ho et al., 2020). For instance, quasi-conservation of a global charge $N$ can be rigorously demonstrated, yielding prethermal regimes protected by emergent U(1) or $\mathbb{Z}_n$ symmetry even when no energy conservation applies at the static Hamiltonian level (Ho et al., 2020).
• Kinetic and semiclassical Boltzmann equations describe the dynamics of slow relaxation and the approach to the prethermal plateau, particularly in systems with dilute quasiparticle excitations or "hard-rod" constraints (Birnkammer et al., 2022).
• Random matrix theory and rigorous ensemble bounds show that prethermalization is typical for a broad class of weakly perturbed integrable systems (Reimann et al., 2019). The evolution of observables can be bounded by

$$\overline{|\Delta(t)|} \leq \frac{\Delta_A}{2}\left[3\sqrt{1-Y(t)} + \sqrt{1-Y(t)+W(t)}\right],$$

where $\Delta_A$ is the spectral range of the observable and $Y(t), W(t)$ capture random-matrix-averaged overlap functions that encode the “quasi-conservation” window.

7. Broader Implications and Outlook

Prethermal phases represent a robust route to realizing nonequilibrium order and transitions outside the constraints of equilibrium statistical mechanics. Their phenomenology—persistent magnetization, nematic or chiral currents, discrete time crystalline oscillations, emergent protection by approximate global symmetries—has direct bearing on experiments with ultracold atoms, trapped ions, superconducting circuits, and classical magnetic systems (Kyprianidis et al., 2021, Pizzi et al., 2021, Ying et al., 2021, Paul et al., 15 Apr 2025).

The controlled appearance of long-lived order in clean, disorder-free systems, the flexibility to engineer prethermal lifetimes via drive frequency or interaction strength, and the connection to constrained dynamics and Hilbert-space fragmentation make prethermal phases a central topic in the understanding and design of dynamical quantum matter.

In summary, the theory and observation of prethermal phases have transformed our conception of nonequilibrium dynamics, revealing regimes where coherent order and symmetry breaking persist far beyond naive expectations, and where the ultimate approach to equilibrium is governed by finely-tuned, system-specific pathways determined by emergent effective Hamiltonians, global symmetries, and the intricate structure of collective excitations.