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REVIEW 3 major objections 4 minor 64 references

Fractionalized Prethermalization in the One-Dimensional Hubbard Model

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A periodically driven one-dimensional Hubbard model can form a quasi-steady state in which charge quasiparticles are hot while spin quasiparticles remain cold, a fractionalized prethermal plateau beyond any single local Floquet Hamiltonian.

desk verdict The fractionalized prethermal plateau in the driven Hubbard model is credible and the drive classification is useful, but the Class (II) lifetime formula in Fig. 1(b) is contradicted by the paper's own Fig. 5(d) fit. read the letter →

arxiv 2502.09708 v3 pith:77HB33T2 submitted 2025-02-13 cond-mat.str-el

Anton Romen , Johannes Knolle , Michael Knap This is my paper · ORCID
classification cond-mat.str-el
keywords fractionalizedprethermalizationHubbardmodelspin-chargeseparationtJFloquetdrivingSchrieffer-Wolfftransformationexactdiagonalizationultracoldatoms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Periodically driven quantum many-body systems usually heat up toward infinite temperature unless the drive is so fast that heating is exponentially suppressed, in which case the system is described by a single local effective Hamiltonian at one temperature. The paper argues that the strongly interacting one-dimensional Hubbard model, whose electrons split into independent spin and charge quasiparticles, breaks this rule: at drive frequencies of order the hopping amplitude, a quasi-steady state forms in which the charge sector is hot while the spin sector stays at low effective temperature. This fractionalized prethermal plateau cannot be described by any single effective Floquet Hamiltonian. The paper classifies drives into three groups, charge-only coupling, asymmetric spin-charge coupling, and strong coupling to both, and shows that the plateau lifetime is set either by quasiparticle decay, scaling as $U^2/t^2$, or by an exponentially long Floquet lifetime. Establishing this in a model naturally realized with ultracold atoms makes fractionalized prethermalization a plausible experimental observable rather than a special feature of one model.

What carries the argument

The central machinery is a time-dependent Schrieffer-Wolff transformation for driven Hubbard models. This transformation removes doublon-holon processes order by order in $t/U$ and yields an effective driven low-energy Hamiltonian; its derivative terms enter only at relative order $\omega/U$, so for $\omega\sim t$ and $U\gg t$ the generator can be approximated as time independent, giving a driven tJ model. Into that effective model the paper inserts the squeezed-space parton representation, where holes become bosonic chargons hopping on the links of a squeezed spin chain and spins become fermionic spinons with Heisenberg exchange that is switched off at occupied links. The separation $H_0+H_{\rm int}$, a non-interacting chargon-and-spinon part plus a static coupling of order $J$, identifies the quasiparticle lifetime via Fermi's golden rule and organizes the drive classification: whether the drive enters through the chargon kinetic term (order $t$), through the spinon exchange term (order $J$), or through both.

What would settle it

Measure the spin-spin correlation decay time $\tau_{\rm th}$ in the driven Hubbard model with a staggered-potential drive as a function of $U/t$ at fixed $\omega=4t$; if the extracted exponent in $\tau_{\rm th}\sim U^\alpha$ deviates from 2 at the largest accessible $U$, or if the effective spin temperature rises on the same timescale as the charge temperature, the fractionalized plateau is not present. A direct check of the machinery would be to solve the full time-dependent Schrieffer-Wolff equation for $S_1(\tau)$ including the $i\partial_\tau S_1$ term at $U=10t$ and verify that the drive-induced spinon coupling stays of order $J$ rather than order $t$.

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Extended reading notes

Core claim

The paper's central claim is that a periodically driven Hubbard model at strong coupling hosts fractionalized prethermalization: for drive frequencies $\omega$ of order the hopping $t$ but large compared to the spin exchange $J=4t^2/U$, the stroboscopic dynamics enter a quasi-steady state with a high effective temperature in the charge sector and a low effective temperature in the spin sector. The lifetime of this plateau is determined by whichever is shorter of two processes: the Fermi-Golden-rule quasiparticle lifetime $\tau_0\sim U^2/t^2$ and the exponentially long lifetime of a Floquet prethermal plateau, $e^{O(\omega U/t^2)}$. Drives that couple only to chargons, for example a staggered potential, produce the FGR-limited plateau; drives that couple strongly to chargons and only weakly to spinons, such as staggered hopping, produce the exponential plateau; drives that couple strongly to both sectors destroy spin-charge separation and thermalize rapidly. The paper derives this structure from the underlying Hubbard model through a time-dependent Schrieffer-Wolff transformation, and shows numerically that the effect survives in the full Hubbard model rather than only in its effective tJ description.

Load-bearing premise

The load-bearing premise is that the perturbative transformation used to derive the effective low-energy model is accurate when truncated at second order in hopping over interaction, and that the neglected frequency-dependent corrections, of order $\omega/U$ for drive frequencies comparable to the hopping, are small enough not to couple the drive directly to the spin degrees of freedom; for the interaction values used in the numerics ($U=10t$ to $40t$), those corrections are only moderately small.

Editorial extensions

If this is right

  • For the staggered-potential drive (Class I), the charge sector reaches an effectively infinite temperature on a timescale of order $10^2/t$, while the spin sector keeps most of its initial correlations up to times of order $U^2/t^2$.
  • For the staggered-hopping drive (Class II), the spin sector exhibits a prethermal plateau with lifetime growing exponentially in $\omega U/t^2$, bounded from above by the quasiparticle lifetime at large $U$.
  • The two sectors have effective temperatures differing by roughly two orders of magnitude throughout the plateau, ruling out a single-temperature Gibbs ensemble of any local effective Floquet Hamiltonian.
  • The same qualitative features appear in numerical simulations of the full driven Hubbard model, with better agreement with the effective tJ model as $U/t$ increases.
  • A drive that couples strongly to both spinons and chargons, such as a next-nearest-neighbor $S^z S^z$ drive, eliminates the plateau: spin and charge heat on the same timescale.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct experimental test is available: in an optical lattice with a superlattice drive, measure the spin correlation lifetime as a function of $U/t$ at fixed drive frequency; a clean $U^2$ scaling identifies a charge-only drive, while a crossover to exponential-in-frequency scaling would reveal a spinon coupling of order $J$.
  • The same spin-charge asymmetry should appear in bosonic Hubbard and tJ models, which the paper mentions as a natural extension; one could probe those with time-dependent density measurements rather than spin-resolved imaging.
  • Because the derivative corrections grow as $\omega/U$, varying the drive frequency toward $U$ at fixed $t/U$ provides a sharp test of the time-dependent Schrieffer-Wolff truncation: the plateau lifetime should drop once the neglected corrections become sizable.
  • The narrow frequency window for the exponential plateau suggests that drive engineering, for instance shaping the pulse to avoid charge-sector prethermalization at higher frequencies, could widen the regime where fractionalized prethermalization is observable.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies a periodically driven one-dimensional hole-doped Hubbard model in the strong-coupling regime. Using a time-dependent Schrieffer-Wolff transformation and exact diagonalization of the low-energy t-J model, the authors argue that at intermediate drive frequencies the system develops a fractionalized prethermal plateau in which the charge sector heats to a high effective temperature while the spin sector remains cold. They classify drives into three classes: drives that couple only to chargons (Class I, lifetime ~ U^2/t^2), drives that couple weakly to spinons (Class II, lifetime min(O(U^2/t^2), e^{O(omega U/t^2)})), and drives that couple strongly to both sectors (Class III, no plateau). They derive the low-energy effective driven Hamiltonian for a staggered-hopping drive, present Hubbard-model exact-diagonalization data that qualitatively match the t-J results, and propose an optical-lattice realization.

Significance. If correct, this work would establish fractionalized prethermalization as a generic phenomenon in one-dimensional spin-charge separated systems, going beyond the Kitaev spin liquid example, and would provide a practical drive classification. Strengths include the transparent squeezed-space parton mapping, the use of multiple diagnostics (kinetic energy, spin correlations, entanglement entropy), a direct Hubbard-model comparison in Fig. 7, and the concrete superlattice protocol in Appendix D; the data and code are promised on Zenodo. However, the most quantitative claim, the U-dependent exponential lifetime of Class (II), is not supported by the shown numerics, and the Schrieffer-Wolff derivative corrections are not fully accounted for; these issues need to be resolved before the central classification can be accepted as stated.

major comments (3)
  1. [Sec. III B, Fig. 5(d)] The claimed Class (II) lifetime is inconsistent with the presented data. The text and Fig. 1(b) give tau_pre ~ e^{O(omega/J)} = e^{O(omega U/t^2)}. With U=26t and J=4t^2/U ~ 0.154t, the exponent is about 6.5 omega, so even at the lowest shown frequency omega=2.4 the predicted prethermal lifetime is e^{15.6} ~ 6 x 10^6, already far above the quasiparticle cutoff U^2/t^2 = 676; the min() formula would then predict tau_th ~ 676, not the factor ~100 growth shown. The fit in Fig. 5(d), labeled ~e^omega, has no U enhancement and is inconsistent with e^{6.5 omega} by many orders of magnitude in the same range. Since only U=26 is simulated, the U dependence that distinguishes e^{omega/t} from e^{omega U/t^2} is not tested. The authors should either provide multi-U data demonstrating the omega/J collapse or revise the lifetime formula in the abstract, Fig. 1(b), and Sec. III B.
  2. [Sec. IV, Eqs. (27)-(37); Appendix C] The treatment of derivative terms in the time-dependent Schrieffer-Wolff expansion is not yet complete. The text first states that i*d_tau S_n gives corrections of relative order O(omega/U) and is negligible for U >> omega, then concludes that corrections to the second-order exchange strength are O(omega^2/U^3). However, Eq. (C13) contains the block-diagonal term H_delta^{(2)} = (1/2)(S_1 d_tau S_1 - (d_tau S_1) S_1), which scales as O(omega t^2/U^2), i.e. O(omega/U) relative to J, and this term is not included in the modified J' expression of Eq. (C15). For the parameters used in the numerics (omega ~ t, U in [10t, 40t]) this is a 10-40% correction to the effective spin-sector coupling, precisely the quantity controlling the Class (I)/(II) distinction. Please clarify whether H_delta^{(2)} contributes to the low-energy spin Hamiltonian and, if it does, include it in the derivation and in the estimate of the prethermal lifetime.
  3. [Sec. III, Eq. (10); Appendix B 2] The lifetime extraction uses the rescaled observable Z_tilde(tau) = Z(tau) - Z(tau=10^5 t) rather than the deviation from the true infinite-temperature value. At large U the late-time value is not close to Tr(Z)/Tr(I), as the authors acknowledge, and the plateau lifetime is defined through decay of this shifted quantity. This post-hoc normalization can bias the extracted tau_th, particularly for the long-plateau regimes central to the U^2 and exponential scaling claims. The L=20 versus L=22 comparison in Fig. 9 is useful, but the Class (II) data in Fig. 5 are for a single U and no such cross-check is shown there; please demonstrate that the reported scalings are insensitive to the choice of late-time reference or to an alternative definition of tau_th.
minor comments (4)
  1. [Fig. 3 caption] The caption contains the typo 'fractionalized perthermal plateau'; it should read 'fractionalized prethermal plateau'.
  2. [Sec. III B and Fig. 5] The notation for time is sometimes confusing: Eq. (5) uses t*tau_0 for the quasiparticle lifetime, while Fig. 5(d) labels the extracted quantity tau_th; stating explicitly that all lifetimes are in units of t^{-1} would improve readability.
  3. [Eq. (2)] The spin-interaction term in the t-J Hamiltonian is written without an explicit projector onto the singly occupied subspace; adding P_0 to that term would make the strong-coupling derivation more transparent.
  4. [Appendix D] The dynamic stabilization s_1(tau) = s_1 + s_2(tau)/2.7 is quoted with a 5% deviation; stating the range of s_2 over which this calibration holds would aid experimental reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central predictions are derived from Fermi's Golden Rule and Floquet prethermalization arguments and checked against exact-diagonalization numerics; the Fig. 5(d) lifetime scaling is a quantitative concern, not a circular reduction.

full rationale

The paper's central derivation starts from the driven Hubbard Hamiltonian and constructs the low-energy effective theory through a time-dependent Schrieffer-Wolff transformation (Sec. IV, Eqs. (25)-(36)), with the O(omega/U) derivative corrections explicitly estimated rather than tuned to reproduce the target plateau. The Class (I) FGR lifetime U^2/t^2 follows from a standard golden-rule estimate of the static spinon-chargon coupling (Eq. (5)) and is compared with, not fitted to, the exact-diagonalization lifetimes (Fig. 3(d)). The Class (II) exponential prethermal lifetime is imported from standard prethermalization theorems (Refs. [17-19,54]) with the spin-drive amplitude O(J) derived from the SW effective Hamiltonian, and the numerical plateau is then checked against this expectation (Fig. 5). Refs. [34] and [46] are self-citations, but they supply only motivation and a rederived squeezed-space formalism; the present model, drive classification, and two-temperature plateau are demonstrated by independent exact-diagonalization data, including the Hubbard-model check of Fig. 7. The e^omega fit in Fig. 5(d) at U=26 is inconsistent with the stated e^{O(omega U/t^2)} enhancement, but this is a quantitative/falsifiability concern about the lifetime formula, not a circularity: the prediction is not definitionally equal to its inputs.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new particles or mediators are introduced. The chargon and spinon are standard partons from Ref. [46]. The central claim rests on the strong-coupling expansion, the squeezed-space exact mapping, Fermi's Golden Rule, and prethermalization theorems.

free parameters (2)
  • Drive amplitude g = 0.324t in Class (I) simulations = 0.324 t
    Chosen by hand as a moderate drive strength; the claimed U^2 scaling is independent of this value to leading order, so it does not bias the central result.
  • Lattice stabilization factor 2.7 in s1(tau) = s1 + s2(tau)/2.7 = 2.7
    Numerically determined in Appendix D to keep hopping and interaction within 5% during superlattice modulation; relevant only to the experimental proposal, not to the central physics claim.
assumptions (6)
  • domain assumption At U >> t and low hole doping, the Hubbard model is equivalent to the tJ-model, neglecting the O(J/8) three-site terms.
    Used in Sec. II Eq. (2) and throughout; standard strong-coupling expansion.
  • domain assumption The squeezed-space parton mapping (bosonic chargon + fermionic spinon) exactly captures the low-energy tJ Hilbert space and separates spin and charge degrees of freedom.
    Taken from Ref. [46]; used in Sec. II and Appendix A. Exact for the tJ-model, approximate for the underlying Hubbard model.
  • standard math Fermi's Golden Rule gives the quasiparticle lifetime t*tau0 ~ t^2/J^2 (Eq. 5).
    Standard perturbation theory with H_int of order J; assumes weak, Markovian coupling.
  • domain assumption A Floquet prethermal plateau exists for the driven spin sector with exponentially long lifetime, per Refs. [17-24,54].
    Invoked in Sec. III B for Class (II); requires a local effective Hamiltonian and drive frequency large relative to the local bandwidth, which is not analyzed for the coupled spinon-chargon system.
  • ad hoc to paper In the time-dependent Schrieffer-Wolff transformation, the derivative term i*∂_τ S_n is of order O(omega/U) relative to leading terms and can be neglected for U >> omega.
    Sec. IV, around Eqs. (30)-(34); a scaling argument specific to this work, whose quantitative validity at omega ~ t and U in [10,40] is not established.
  • domain assumption The non-interacting quasiparticle description H0 = T + D + S (independent chargon and spinon) remains valid up to the quasiparticle lifetime.
    Used in Sec. II to define the FGR decay; the central claim requires that spin-charge separation survives for times much longer than the chargon thermalization time.

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Pith. "Pith review of Fractionalized Prethermalization in the One-Dimensional Hubbard Model." pith.science (2026). https://pith.science/paper/77HB33T2

@misc{pith2026250209708,
  author       = {Pith},
  title        = {Pith review of: Fractionalized Prethermalization in the One-Dimensional Hubbard Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/77HB33T2}},
  note         = {Machine review of arXiv:2502.09708}
}
read the original abstract

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 quasi-stationary state that is not captured by this paradigm. Instead distinct types of fractionalized excitations are characterized by vastly different temperatures-a phenomenon dubbed "fractionalized prethermalization". 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 quasi-steady 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 towards an experimental realization of this phenomenon in quantum simulation platforms.

Figures

Figures reproduced from arXiv: 2502.09708 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
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Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

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