Quantitative theory of fractionalized prethermalization in the driven one-dimensional Hubbard model: lifetime crossover, drive-class boundaries, and a controlled time-dependent Schrieffer-Wolff expansion

Romen, Knolle, and Knap [Phys. Rev. B 112, 195124 (2025)] showed that the strongly coupled, weakly hole-doped one-dimensional Hubbard model driven at intermediate frequencies hosts a fractionalized prethermal quasi-steady state: the charge sector heats rapidly to a high effective temperature while the spin sector remains in a low-temperature prethermal state, with the plateau lifetime governed by the competition between a Fermi-golden-rule quasiparticle lifetime $\tau_0$ with $t\,\tau_0 \sim U^2/t^2$ and an exponential-in-frequency prethermal lifetime $\tau_{\mathrm{pre}}$ of order $e^{\omega U/t^2}$ . They classified drives into three classes according to the existence and lifetime of the fractional plateau, derived via a time-dependent Schrieffer-Wolff transformation in which time-derivative terms are dropped and corrections only estimated as $\mathcal{O}(\omega/U)$ . The paper leaves the quantitative theory open: no analytic derivation of the lifetime crossover including prefactors, no quantitative boundaries between the drive classes as functions of the microscopic drive form, amplitude, frequency, filling, and interaction strength, and no rigorous error bounds for the Schrieffer-Wolff scheme. The problem of turning this qualitative picture into a controlled quantitative theory remains open.

Background

A quantum many-body system driven by a periodic force (a Floquet system) generically absorbs energy from the drive and heats toward an infinite-temperature state. A standard escape route is prethermalization: at drive frequencies large compared to the internal local energy scales, heating rates are exponentially suppressed in the frequency and the system lingers for exponentially long times in a quasi-steady Gibbs state of an effective Hamiltonian computed by a high-frequency (Floquet-Magnus) expansion; rigorous versions of this statement are known for bounded quantum systems (Bukov, D'Alessio, and Polkovnikov, Adv. Phys. 64, 139 (2015); Abanin, De Roeck, Ho, and Huveneers, Commun. Math. Phys. 354, 809 (2017)). Recently this paradigm was challenged in a driven Kitaev honeycomb spin liquid, whose fractionalized excitations (matter fermions and static fluxes) acquired vastly different effective temperatures that cannot be captured by a single-temperature effective Hamiltonian — a phenomenon dubbed fractionalized prethermalization (Jin, Knolle, and Knap, Phys. Rev. Lett. 130, 226701 (2023)). Because the Kitaev model has an exactly solvable sector structure, it was unclear whether other fractionalized systems behave similarly. Romen, Knolle, and Knap [Phys. Rev. B 112, 195124 (2025); arXiv:2502.09708] addressed this for the paradigmatic case of spin-charge separation in one dimension: in the strongly coupled ( $U \gg t$ ), weakly hole-doped (a density slightly below one electron per site) Hubbard chain, the electron fractionalizes into a fast chargon (a charge defect hopping with amplitude $t$ , where $t$ is the nearest-neighbor hopping) and a slow spinon (a spin excitation moving at the superexchange scale $J = 4t^2/U$ , where $U$ is the on-site repulsion), described at low energies by a tJ model on a constrained Hilbert space (the squeezed space of the spin-charge separated degrees of freedom). Driving at intermediate frequencies, of order the hopping $t$ but large compared to $J$ , the charge sector heats quickly while the spin sector remains prethermal, producing a two-temperature fractional quasi-steady state. The paper classifies drives — periodic perturbations of the form $g f(\tau) \hat{V}$ of amplitude $g$ and frequency $\omega$ added to the tJ Hamiltonian — into three classes: Class (I), coupling only to the charge sector, yields a fractional plateau whose lifetime is limited by the quasiparticle lifetime $\tau_0$ (with $t\,\tau_0 \sim U^2/t^2$ from Fermi's golden rule (FGR)); Class (II), coupling asymmetrically (strongly to charge, of order $J$ to spin), yields a plateau whose lifetime is the smaller of $\tau_0$ and the exponential prethermal lifetime $\tau_{\mathrm{pre}}$ of the spin sector, of order $e^{\omega U/t^2}$ , i.e. exponential in the drive frequency measured in units of the spinon scale $J$ ; Class (III), coupling strongly to both quasiparticles, destroys spin-charge separation and the system heats rapidly with no fractional plateau. Numerically, the three classes are realized by three concrete drives: a staggered potential (Class I), staggered hopping combined with a Heisenberg drive (Class II), and a drive coupling strongly to the spins (Class III). The effective theory is derived by a time-dependent variant of the Schrieffer-Wolff (SW) transformation — the standard block-diagonalizing perturbative rotation for quantum many-body Hamiltonians (Bravyi, DiVincenzo, and Loss, Ann. Phys. 326, 2793 (2011)); a related construction for periodically driven strongly correlated systems appears in Bukov, Kolodrubetz, and Polkovnikov, Phys. Rev. Lett. 116, 125301 (2016). In the time-dependent setting the SW generators satisfy first-order inhomogeneous linear differential equations rather than algebraic recursion relations; Romen et al. drop the time-derivative terms at leading order and estimate the resulting corrections only at relative order $\mathcal{O}(\omega/U)$ . Their numerical evidence comes from exact diagonalization of systems with at most $L = 22$ sites and 2-4 holes, with acknowledged finite-size offsets and ergodicity-restoration issues. The phenomenon is within reach of ultracold-atom quantum simulators, where Floquet prethermalization has already been observed in a Bose-Hubbard system (Rubio-Abadal et al., Phys. Rev. X 10, 021044 (2020)) and spin-charge deconfinement has been observed in Hubbard chains (Vijayan et al., Science 367, 186 (2020)); the paper's numerical data and code are deposited at Zenodo (DOI 10.5281/zenodo.14794905) with restricted access, available from the authors upon reasonable request.

Problem Statement

Turn the qualitative picture of fractionalized prethermalization in the driven one-dimensional Hubbard model (Romen, Knolle, and Knap, Phys. Rev. B 112, 195124 (2025)) into a controlled quantitative theory. The single overarching objective is an analytically derived, error-controlled account of the fractional prethermal plateau — its existence, its lifetime, and its boundaries in parameter space — replacing the current combination of leading-order heuristics and finite-size numerics. Concretely, for the strongly coupled ( $U \gg t$ ), weakly hole-doped one-dimensional Hubbard model driven periodically at intermediate frequencies ( $\omega$ of order $t$ , large compared to the superexchange $J = 4t^2/U$ ):

Derive the lifetime competition principle with prefactors: establish analytically that the lifetime of the fractional prethermal plateau is governed by the smaller of the quasiparticle lifetime $\tau_0$ (Fermi-golden-rule decay of the spin-charge separated quasiparticles, with $t\,\tau_0 \sim U^2/t^2$ ) and the exponentially long prethermal lifetime $\tau_{\mathrm{pre}}$ of the spin sector (of order $e^{\omega U/t^2}$ ), including the leading numerical prefactors of both times and the crossover condition separating the FGR-limited from the exponentially limited regime, together with the parameter window in which the exponential regime is observable before quasiparticle decay dominates.
Determine the boundaries between the three drive classes quantitatively: Class (I) drives coupling only to the charge sector (plateau lifetime limited by $\tau_0$ ), Class (II) drives coupling asymmetrically (plateau lifetime $\min(\tau_0, \tau_{\mathrm{pre}})$ ), and Class (III) drives coupling strongly to both sectors (breakdown of spin-charge separation, no fractional plateau). This requires explicit criteria, in terms of the microscopic drive form and its matrix elements between the charge and spin sectors, amplitude $g$ , frequency $\omega$ , filling, and $U/t$ , that locate the transitions between these classes.
Put the underlying time-dependent Schrieffer-Wolff expansion on controlled footing: retain and treat the time-derivative terms to the needed higher order (they are currently dropped, with corrections only estimated as $\mathcal{O}(\omega/U)$ ), and obtain rigorous error bounds for finite $U$ , $\omega$ , and evolution times that establish the radius of validity of the effective theory from which the class structure and lifetime formulas follow.

Deliverables (1)-(3) are jointly supporting: the class boundaries of (2) and the crossover formula of (1) are outputs of the effective theory whose control is deliverable (3), so an answer that resolves the lifetime principle while leaving the class boundaries merely qualitative, or that derives formulas without controlling the approximation on which they rest, does not complete the objective. The paper's published exact-diagonalization results (lifetime data for $L = 20$ - $22$ sites and $10 \le U \le 40$ ; the underlying data and code are deposited at Zenodo, DOI 10.5281/zenodo.14794905, with restricted access available from the authors upon reasonable request) are the natural numerical reference point against which such a theory can be compared. The broader questions the source also signals — which other fractionalized systems (for example bosonic Hubbard or tJ variants, or higher-dimensional systems) exhibit fractionalized prethermalization, and how the phenomenon could be realized experimentally in noisy quantum-simulation platforms — are natural extensions of this theory but are not part of what completes it.

References

A. Romen, J. Knolle, and M. Knap, "Fractionalized prethermalization in the one-dimensional Hubbard model", Phys. Rev. B 112, 195124 (2025), DOI 10.1103/qlky-lzx6, arXiv:2502.09708, https://arxiv.org/abs/2502.09708
H.-K. Jin, J. Knolle, and M. Knap, "Fractionalized Prethermalization in a Driven Quantum Spin Liquid", Phys. Rev. Lett. 130, 226701 (2023), DOI 10.1103/PhysRevLett.130.226701, arXiv:2211.10453, https://arxiv.org/abs/2211.10453
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A. Romen, J. Knolle, and M. Knap, data and simulation code for "Fractionalized Prethermalization in the One-Dimensional Hubbard Model", Zenodo (restricted access, available from the authors upon reasonable request), DOI 10.5281/zenodo.14794905, https://zenodo.org/records/14794905