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REVIEW 3 major objections 3 minor 71 references

Observation of slow relaxation due to Hilbert space fragmentation in strongly interacting Bose-Hubbard chains

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Pith's one-line read A 1D Bose-Hubbard chain with strong interactions and no disorder or tilt exhibits slow relaxation because the numbers of doublons and singlons are emergent conserved quantities that fragment the Hilbert space.

desk verdict A careful experiment with solid controls, but the claim that the slow relaxation indicates Hilbert space fragmentation is not uniquely supported; ordinary large-U dynamics with singlon-assisted transport explains the data just as well. read the letter →

arxiv 2502.02959 v2 pith:2A4SL34P submitted 2025-02-05 cond-mat.quant-gas cond-mat.stat-mechquant-ph

Kantaro Honda , Yosuke Takasu , Shimpei Goto , Hironori Kazuta , Masaya Kunimi , Ippei Danshita , Yoshiro Takahashi This is my paper · ORCID
classification cond-mat.quant-gascond-mat.stat-mechquant-ph
keywords HilbertspacefragmentationBose-Hubbardmodelquantumthermalizationdoubloncharge-densitywaveultracoldatomsopticallatticenonergodicdynamicsemergentconservedquantities
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

The paper reports an ultracold-atom experiment on a one-dimensional Bose-Hubbard chain with neither disorder nor tilt, starting from a period-two charge-density wave of doublons (atoms paired on every other site). In the strongly interacting regime, it argues, the numbers of doublons and singly occupied sites become emergent conserved quantities, so the Hilbert space fragments into an exponentially large number of disconnected sectors and relaxation slows down. The authors observe that the doublon imbalance stays far from zero over about ten tunneling times while the singlon imbalance relaxes quickly, and that both particle-number fractions remain essentially constant. They further find that removing some of the singlons makes the doublon relaxation even slower, as expected when singlons provide the only zero-energy tunneling channel. If correct, the experiment shows that strong interactions alone, in a clean untilted system, can produce nonergodic dynamics of the Hilbert-space-fragmentation type.

What carries the argument

The load-bearing object is the one-dimensional Bose-Hubbard Hamiltonian $$\hat H = -J \sum_{\langle i,j\rangle} \hat a_i^\dagger \hat a_j + \frac{U}{2} \sum_i \hat n_i(\hat n_i-1) + \sum_i V_i \hat n_i$$ with a weak parabolic trap $V_i = \Omega[i-(M+1)/2]^2$ and no tilt. Around this Hamiltonian the paper organizes a state-preparation sequence that produces the doublon CDW, a singlon/doublon-resolved measurement based on photoassociation and band mapping, and an effective description in which doublons hop with amplitude $J_{\mathrm{eff}} = 2J^2/U$. In the large-$U$ limit the doublon-holon system is mapped to spin-1/2 particles with a nearest-neighbor interaction, which is used to explain why doublons relax even more slowly than non-interacting hard-core bosons; time-evolving block decimation with matrix-product states supplies the numerical reference curves.

What would settle it

A decisive check is to measure $n_D$ and $I_D$ at holding times beyond about 20 tunneling times while suppressing transverse hopping with a deeper lattice along the perpendicular directions; if $n_D$ decays measurably or $I_D$ crosses zero within that window, the conserved quantities are only transient and the slowdown is a prethermal effect rather than Hilbert space fragmentation. Alternatively, a numerical count of Krylov sectors for the Hamiltonian in Eq. (2) at $U/J \simeq 67$ would settle the mechanism: if the number of sectors does not grow exponentially with chain length, the parabolic trap, not fragmentation, is responsible for the slow relaxation.

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Extended reading notes

Core claim

The central claim is that in the one-dimensional Bose-Hubbard model at $U/J \gg 1$, the total numbers of doublons and singlons are emergent conserved quantities, and these conserved quantities fragment the Hilbert space into an exponentially large number of Krylov subsectors (the disjoint state spaces generated by repeatedly applying the Hamiltonian). Starting from a period-two doublon charge-density wave $|\cdots2020\cdots\rangle$ (with a 25--30% singlon admixture), the dynamics should therefore be restricted to a tiny fragment of the Hilbert space: doublons cannot dissociate into singlons because the process $|2,0\rangle \leftrightarrow |1,1\rangle$ costs energy $U$, while singlons can still tunnel freely. The experiment reports that the doublon imbalance relaxes slowly and remains nonzero for hold times up to roughly ten tunneling times, whereas the singlon imbalance relaxes rapidly; concurrently $n_D$ and $n_S$ stay almost constant. Removing part of the initial singlon population slows the doublon relaxation further, confirming that singlons act as the main agent of equilibration. The paper takes this as experimental confirmation that the conserved quantities responsible for Hilbert space fragmentation exist in a disorder-free, untilted bosonic system.

Load-bearing premise

The experiment assumes that the superlattice merging procedure actually creates a period-two doublon charge-density wave and that photoassociation-based detection cleanly separates singlons from doublons; if either fails, the apparent conservation of n_D and n_S would not be evidence for Hilbert space fragmentation.

Editorial extensions

If this is right

  • In a clean, untilted bosonic chain, strong interactions alone are enough to keep the system out of thermal equilibrium for tens of tunneling times, so Hilbert space fragmentation is a real physical mechanism rather than a tilt- or disorder-induced artifact.
  • The conserved doublon and singlon numbers protect the doublon CDW: dissociation $|2,0\rangle \to |1,1\rangle$ is kinematically forbidden at large $U/J$.
  • The relaxation of the doublon imbalance is controlled by the singlon population, so reducing singlon contamination makes the nonergodic behavior sharper and lengthens the relaxation time.
  • The singlon/doublon-resolved imbalance protocol can be applied to other composite-particle dynamics in optical lattices, including pair superfluids, $\eta$-pairing states, and bosonic many-body scars.
  • Because the kinetic constraint comes from the interaction rather than the trap, the fragmentation picture is not tied to a particular system size or to the specific parabolic potential used in the experiment.

Reading between the lines

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

  • If the fragmentation is genuine, the asymptotic doublon imbalance in the pure-doublon limit ($n_S \to 0$, $U/J \to \infty$) should saturate at a nonzero value controlled by the parabolic trap; measuring the imbalance at much longer hold times under stronger transverse confinement would test this directly.
  • The singlon-assisted relaxation suggests a defect-controlled picture in which the equilibration time scales with the inverse singlon density; plotting the doublon relaxation time against $n_S$ across the accessible range could expose a power law and distinguish fragmentation from a simple prethermal plateau.
  • The same PA-based species-resolved imbalance technique could be carried over to tilted Fermi-Hubbard chains to separate the conserved dipole-moment sector (tilt-induced) from interaction-induced doublon-number sectors, since both give slow relaxation but with different selection rules.
  • A cleaner test of Hilbert space fragmentation would be to analyse the level statistics or sector decomposition of the Hamiltonian in Eq. (2) numerically: if the number of Krylov sectors grows only polynomially with chain length, the slow relaxation is more naturally attributed to the trap than to fragmentation.
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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 / 3 minor

Summary. The manuscript reports ultracold-atom experiments on strongly interacting 174Yb bosons in 1D optical lattices. Starting from an approximate period-two charge-density wave of doublons (with 25-30% singlon contamination), the authors suddenly quench the lattice depth and measure, using photoassociation-based removal, the time-resolved imbalances of doublons and singlons separately. They observe that the doublon imbalance relaxes slowly and the singlon imbalance rapidly, that the fractions of atoms in doublons and singlons stay nearly constant, that doublon relaxation slows with increasing U/J, and that partial removal of singlons slows the doublon relaxation. TEBD calculations of the Bose-Hubbard model with a parabolic trap reproduce the main trends. The authors interpret the near-conservation of doublon and singlon numbers as evidence for Hilbert-space fragmentation and conclude that this mechanism causes the observed slow relaxation.

Significance. If the attribution to Hilbert-space fragmentation were established, this would be a noteworthy experimental demonstration of an emergent conserved quantity in a disorder-free and untilted 1D Bose-Hubbard system. The paper has concrete strengths: the observables are direct and species-resolved, with error bars based on multiple scans; the comparison between doublon-CDW and singlon-CDW initial states is a useful control; the U/J scan and partial-singlon-removal experiment provide parameter dependence in the expected direction; and the TEBD comparisons give an honest account of quantitative disagreement. However, the specific signatures that would distinguish Hilbert-space fragmentation from generic large-U dynamics are not isolated, and the main interpretive claim is therefore underdetermined by the presented data.

major comments (3)
  1. [Results: Typical quench dynamics, Fig. 1F and Fig. 2E] The near-conservation of nD and nS is presented as the key evidence for HSF ('strongly supports the occurrence of HSF'). This inference is not valid: for the Hamiltonian in Eq. (2), every process that changes nD or nS, such as |20> <-> |11>, is off-resonant by an energy U. The probability of such a process during one tunneling attempt is of order (J/U)^2, so after tJ/hbar = 10 at U/J = 67 the expected change in nD and nS is about 10(J/U)^2, i.e., 2x10^-3, far below the experimental resolution. A large-U Bose-Hubbard model with no fragmentation whatsoever would show exactly the same flat nD and nS curves. The measurement of nD and nS conservation is therefore necessary but not sufficient for the HSF claim.
  2. [Results: Competition among doublon-doublon interactions, doublon-singlon interactions, and a parabolic trap, Figs.] The relaxation of the doublon imbalance that the authors attribute to fragmentation is, on their own analysis, driven by the resonant doublon-singlon swap |2,1> <-> |1,2> (Fig. 4A), a process that conserves nD and nS exactly and has amplitude J rather than J^2/U. Since the initial state contains 25-30% singlons, this process allows doublons to be transported and ID to decay on a few-hbar/J time scale, as the authors use it to explain why the observed doublon dynamics is faster than the noninteracting hardcore-boson prediction in Fig. 3C. Thus the simultaneous observations that nD and nS are flat while ID decays slowly are fully consistent with ordinary large-U dynamics with singlon-assisted transport; they do not single out fragmentation. A fragmentation-specific observable, such as the conservation of the number of doublons in each interval between holes, or a plateau of ID at times long compared with all second-order processes, would be required.
  3. [Discussion and observation window] The observation window ends at tJ/hbar = 10, which is comparable to or shorter than the transport time of a singlon across the roughly 25-site chains and much shorter than the second-order doublon-hopping time hbar/(2J^2/U) at the largest U/J. The data show a slow decay of ID but no clear separation of time scales and no long-time plateau that would indicate restriction to a Krylov sector. Without a longer-time measurement or an order-sensitive probe, the observed slow relaxation can be accounted for by the small effective doublon mobility and the doublon-doublon interaction in Eq. (8), both of which are generic strong-coupling effects rather than fragmentation signatures. The claim in the Discussion that the inter-tube hopping is the limiting time scale should be supported by an explicit estimate of the expected fragmentation plateau value and relaxation time, and why the present window suffices.
minor comments (3)
  1. [Fig. 1D caption and main text] The caption of Fig. 1D states U/J = 52, while the main text description of the same panel states U/J = 67; please correct this inconsistency.
  2. [Eqs. (5) and (6)] The definitions nS = Nw/PA / Nw/oPA and nD = (Nw/oPA - Nw/PA) / Nw/oPA are fractions of the total atom number, not absolute numbers; the text sometimes refers to them as the 'numbers' of singlons and doublons, which is imprecise.
  3. [Supplementary Sec. S5] The initial-state simulation protocol determines beta and mu by matching Ntot and Ndoublon for the CDW(d) case, but for the CDW(s) case beta is simply assumed to be betaER = 20 rather than determined from data; the sensitivity of the TEBD comparisons to this assumption should be stated.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the conserved fractions and slow relaxation are directly measured; the coauthored prediction of Ref. [54] is independently tested rather than used to force the result.

full rationale

Walking the derivation chain, the central claims are experimental measurements, not outputs of a fitted or self-referential model. The singlon/doublon fractions are defined by direct photoassociation-resolved atom counts, Eqs. (5)-(6), and the imbalances are defined by site- and band-mapped atom numbers, Eqs. (3)-(4); the near-conservation of nD and nS and the slow doublon relaxation reported in Figs. 1E-F and 2 are raw data. The TEBD calculations (Sec. 'Comparison between experiments and numerical calculations' and Sec. S.5) evolve the first-principles Bose-Hubbard Hamiltonian Eq. (2) with parameters obtained from Wannier functions and the measured scattering length (Sec. S.3). The only calibrated inputs are the temperature and central chemical potential of the initial Mott state, fixed by the measured total atom number and initial doublon number via Eqs. (S13)-(S14); these do not encode the predicted imbalance curves or the conserved-fraction result. The theoretical expectation of nonergodic doublon-CDW dynamics is cited to Ref. [54], whose authors (Kunimi and Danshita) are coauthors of this work; however, that reference is a prior prediction being tested by an independent experiment, and the HSF concept itself is anchored in external references (Refs. [43-50]). The skeptical objection that flat nD/nS over about 10 hbar/J is also compatible with ordinary large-U dynamics is a legitimate concern about discriminative power, but it is not a circular reduction: the paper does not fit the conserved fractions and then present the fit as a prediction. No equation or parameter in the paper makes the claimed conclusion equivalent to its input by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on measured imbalance and number fractions, which do not require free parameters. For the theory comparison, the initial state of the TEBD is fixed by experimentally measured total atom number and doublon number, so beta and mu_ctr are calibrated inputs rather than target-fitting parameters. The main domain assumptions are the validity of the 1D Bose-Hubbard Hamiltonian, the PA-based number resolution, and the theory of emergent conserved quantities from Ref. [54].

free parameters (2)
  • Inverse temperature beta/E_R = 20.1 (for rho_CDW(d,s) case)
    Determined by solving Eqs. (S13)-(S14) with experimentally measured Ntot and Ndoublon; it calibrates the initial state of the TEBD simulation, not the target dynamics.
  • Chemical potential mu_ctr/E_R = 0.326 (for rho_CDW(d,s) case)
    Same fitting procedure as beta; fixes the local density profile used for initial-state sampling.
assumptions (4)
  • domain assumption The system is described by the 1D Bose-Hubbard model of Eq. (2) with a parabolic trap and negligible coupling to other chains during the observation window.
    Used throughout; the paper notes the perpendicular tunneling time exceeds the maximum hold time by a factor of about 3.5 (see Results and Materials and Methods).
  • domain assumption In the U >> J regime the numbers of doublons and singlons are emergent conserved quantities that fragment the Hilbert space, as predicted in Ref. [54].
    Taken from the authors' earlier theoretical work; the interpretation of the measured conservation as HSF depends on this.
  • domain assumption The photoassociation (PA) resonance selectively removes doublons without affecting singlons, and the band-mapping and site-mapping procedure faithfully converts site occupancies to momentum bands.
    Used to define nD, nS and the imbalances; see Sec. 'Atom number imbalance measurement' and Sec. S4.
  • domain assumption The initial state preparation via the superlattice produces the period-two CDW of doublons with single-site occupancy defects, and the temperature and chemical potential of the intermediate Mott state are inferred from a thermal model.
    Secs. S1 and S5; the preparation is calibrated by the measured singlon fraction and initial imbalance.

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Pith. "Pith review of Observation of slow relaxation due to Hilbert space fragmentation in strongly interacting Bose-Hubbard chains." pith.science (2026). https://pith.science/paper/2A4SL34P

@misc{pith2026250202959,
  author       = {Pith},
  title        = {Pith review of: Observation of slow relaxation due to Hilbert space fragmentation in strongly interacting Bose-Hubbard chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2A4SL34P}},
  note         = {Machine review of arXiv:2502.02959}
}
read the original abstract

While isolated quantum systems generally thermalize after long-time evolution, there are several exceptions defying thermalization. A notable mechanism of such nonergodicity is the Hilbert space fragmentation (HSF), where the Hamiltonian matrix splits into an exponentially large number of sectors due to the presence of nontrivial conserved quantities. Using ultracold gases, here we experimentally investigate the one-dimensional Bose-Hubbard system with neither disorder nor tilt potential, which has been predicted to exhibit HSF caused by a strong interatomic interaction. Specifically, we analyze far-from-equilibrium dynamics starting from a charge-density wave of doublons (atoms in doubly occupied sites) in a singlon and doublon-resolved manner to reveal a slowing-down of the relaxation in a strongly interacting regime. We find that the numbers of singlons and doublons are conserved during the dynamics, indicating HSF as a mechanism of the observed slow relaxation. Our results provide an experimental confirmation of the conserved quantities responsible for HSF.

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