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Exponentially slow thermalization in 1D fragmented dynamics

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Pith's one-line read The paper argues that 1D constrained dynamics with exponentially fragmented Hilbert space thermalize in time exponential in system size when touched by a boundary bath, and verifies the claim across several model classes.

desk verdict A genuinely useful conjecture-plus-evidence paper: clean taxonomy, honest gaps, but the broad claim leans on a sketched proof for hyperbolic groups. read the letter →

arxiv 2501.13930 v1 pith:6DWPENPS submitted 2025-01-23 quant-ph cond-mat.stat-mechcond-mat.str-elmath.GR

Cheng Wang , Shankar Balasubramanian , Yiqiu Han , Ethan Lake , Xiao Chen , Zhi-Cheng Yang This is my paper · ORCID
classification quant-phcond-mat.stat-mechcond-mat.str-elmath.GR MSC 05C8120F6782C10
keywords HilbertspacefragmentationthermalizationtimeKrylovgraphconductanceboundarydepolarizingnoiseconstraineddynamicshyperbolicgroupsheatkernelmeasure
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

This paper asks when a 1D system whose dynamics is chopped up by local constraints can still thermalize if one end is touched by an infinite-temperature bath. It argues that if the constraints fragment the Hilbert space exponentially—that is, if the single largest component of states occupies only an exponentially small fraction of the whole space, $|K_{\max}|/|\mathcal{H}| = O(e^{-L})$—then the thermalization time is either infinite or grows exponentially with system size. The authors verify this for several representative models (a charge-biased 'breakdown' chain, the $tJ_z$ model, and a dipole-conserving fracton chain) and prove it for a large class of constraints that come from group multiplication rules. If the conjecture is right, a hard dynamical question—how long a constrained system remembers its initial state—is answered by a simple static count: the fraction of Hilbert space in the largest sector.

What carries the argument

The load-bearing object is the Krylov graph $G_K$, whose vertices are Krylov sectors and whose edges are connections generated by the boundary depolarizing noise; its conductance $\Phi(G_K)$ is the smallest, over all sector sets of at most half the total weight, of the probability of leaving the set in one step. A standard conductance lower bound converts $\Phi$ into a lower bound on $t_{\mathrm{th}}$. For group-based constraints, the relevant graph is a Cayley graph carrying an $L$-step random-walk (heat-kernel) measure, and the proof works by showing that for hyperbolic groups one can cut a ball of the Cayley graph with a shell whose heat-kernel measure is exponentially small, using the finite fractal dimension of the boundary at infinity and concentration of random walks on a sphere. The class 0/I/II taxonomy (never ergodic / exponentially long diameter / short diameter with severe bottlenecks) organizes the three ways conductance can be exponentially small.

What would settle it

Take any 1D constrained model whose largest Krylov sector is an $O(e^{-L})$ fraction of Hilbert space, couple one boundary site to maximally depolarizing noise, and compute or simulate the mixing time of the induced Markov chain for increasing $L$; if $t_{\mathrm{th}}$ grows only polynomially in $L$ for any such model, the conjecture is false.

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Extended reading notes

Core claim

The paper's central claim is the exponential-fragmentation conjecture. In plain terms: let $\mathrm{Dyn}$ be constrained 1D dynamics on $L$ sites with maximally depolarizing noise on one boundary site, and let $|K_{\max}|/|\mathcal{H}|$ be the fraction of Hilbert space in the largest Krylov sector. If this ratio is $O(e^{-L})$, then the conductance of the coarse-grained Krylov graph is $O(e^{-L})$, and by the standard conductance lower bound the thermalization time is $t_{\mathrm{th}} = \Omega(e^{L})$. The paper proves this for all constraints arising from multiplication laws of hyperbolic groups, using the geometry of the boundary at infinity and random-walk heat kernels, and it verifies the bound by direct computation for the spin-1 breakdown model (class I, exponentially large Krylov-graph diameter), the $tJ_z$ model (class II, tree bottleneck), and the range-three dipole-conserving model (class II, real-space bottlenecks). It also shows that a general proof is equivalent to a quantitative refinement of an expander-graph conjecture, and notes that the converse is false: slow thermalization can occur without exponential fragmentation.

Load-bearing premise

The lower bound assumes that the boundary depolarizing channel is the fastest possible constraint-breaking perturbation and that the bulk dynamics can be idealized as instantly mixing within each sector; if a real Hamiltonian has energy conservation, a non-uniform stationary state, or a weaker or less frequent bath, the bottleneck-based bound need not apply.

Editorial extensions

If this is right

  • If the conjecture is correct, the static ratio $|K_{\max}|/|\mathcal{H}|$ becomes a universal predictor: exponential fragmentation immediately gives an exponential lower bound on the thermalization time under boundary noise.
  • Boundary conditions matter: for the $tJ_z$ model one open end gives $t_{\mathrm{th}} \ge (3/8)(3^L-1)$, while baths at both ends relax magnetization in roughly $L^{3.5}$ steps.
  • The range-three dipole-conserving model has real-space frozen regions that make slow thermalization robust even to baths at both boundaries, unlike the pair-flip and $tJ_z$ models.
  • Subsystem thermalization, quantified by the ergodicity length, varies wildly among exponentially fragmented models: infinite for the breakdown and range-three dipole models and exponential for $tJ_z$, so boundary noise is not always a faithful proxy for subsystem dynamics.
  • Because random group presentations are hyperbolic with high probability, the proof for hyperbolic groups covers generic group-based constraints.

Reading between the lines

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

  • One could use the conjecture as a screening tool: compute only the sector-size spectrum of a constrained model and, if it is exponentially fragmented, skip expensive dynamical simulations and expect exponentially slow thermalization under boundary noise.
  • The converse is likely false beyond the paper's impurity example: polynomially fragmented models with deep internal structure may also relax exponentially slowly, so the conjecture should be read as sufficient, not necessary.
  • Testing the quantitative heat-kernel conjecture on groups whose random-walk return probability decays subexponentially would map the boundary between polynomial and exponential thermalization; a polynomial conductance there would disprove the refined conjecture.
  • For Hamiltonian dynamics with energy conservation, the stationary measure is no longer uniform; adapting the argument to non-uniform fixed points would either extend the conjecture to closed quantum systems or reveal a genuine loophole.
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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

4 major / 4 minor

Summary. The paper studies 1D constrained quantum/classical dynamics coupled to maximally depolarizing noise on a boundary site. Its central claim is the Conjecture of Sec. III B: if the dynamics is exponentially fragmented, i.e. |Kmax|/|H| = O(exp(-L)), then the thermalization time satisfies t_th = Ω(exp(L)), equivalently the Krylov-graph conductance obeys Φ(GK) = O(exp(-L)). The authors prove this statement for the spin-1 breakdown model (Class I) and the tJz model (Class II), give heuristic and numerical evidence for the range-three dipole-conserving model, and relate a general proof to a quantitative version of Benjamini's conjecture on heat-kernel-weighted expanders. They also present an informal proof for dynamics based on hyperbolic groups, supported by the Bonk-Schramm embedding theorem and random-walk concentration estimates. The final sections discuss fragmentation transitions, the East model, and several open problems.

Significance. If the central conjecture is correct, it would establish a striking and practically useful principle: a purely static quantity, the fractional size of the largest Krylov sector, universally controls the dynamical thermalization time under local boundary noise. The explicit Cheeger-based lower bounds for the tJz and breakdown models are clean, parameter-free, and constitute genuine progress; the numerical verification of the predicted exponential relaxation in these models is convincing. The reformulation in terms of heat-kernel-weighted conductance and Benjamini's conjecture is insightful and connects the physics problem to an active area of geometric group theory. However, the broadest claims currently rest on an informal theorem and on additional conjectures, so the paper should be read as strong evidence for, rather than a proof of, the general statement.

major comments (4)
  1. [§IV C 1 and Abstract] The abstract states that the conjecture 'holds for a wide range of dynamical constraints, including dipole-conserving dynamics,' but for the range-three dipole model the paper explicitly says: 'While an analytical bound on Φ(GK) and hence t_th has eluded us so far' and then provides only a real-space heuristic and numerical data. As written, this model is suggestive evidence, not a proof. Please either supply an analytic conductance bound for this model or revise the abstract and Sec. IV C to present it as a supported conjecture rather than an established case.
  2. [§V C and App. C.1] Theorem 4 is labeled 'Informal,' and the proof sketch contains a load-bearing step that is not justified. In the proof of Theorem 11, the set S_{m*} ∪ G_{n*} is asserted to 'naturally separate' X ∩ B(L) into two pieces; this is a nontrivial topological claim about cones in arbitrary hyperbolic groups and could fail for Gromov boundaries that are not simple Cantor sets. The measure estimates also rely on an entropy-partition argument with an unspecified constant K, and the Markov-inequality step does not quantify the exceptional values of m. Since this theorem is the main evidence for the broad class of group-based dynamics, the informal status is a genuine gap. The theorem should either be completed with a rigorous separator construction, or the corresponding claims in the abstract and introduction should be explicitly downgraded to heuristic evidence.
  3. [§II B, §III A, and App. A] The lower-bound strategy replaces the bulk dynamics by instantaneous depolarization inside each Krylov sector (Eq. 6) and then uses Lemma 2, which applies only to bistochastic transition matrices with uniform stationary distribution. For Hamiltonian dynamics with energy conservation, or for any dynamics with additional conserved quantities, the stationary distribution is not uniform, and the definition of t_th in Eq. (4) itself requires modification, as the paper acknowledges in Sec. II B. The statement that bounds in the depolarizing model 'also provide bounds' in more structured settings is an expectation, not a proof. Please either restrict the main conjecture to the circuit-averaged setting with a maximally mixed fixed point or provide a concrete reduction argument for non-uniform stationary measures.
  4. [§V B and Conjecture 2] The route to a general proof of the main conjecture is explicitly contingent on the quantitative Benjamini conjecture (Conjecture 2), which is open. Theorem 2 of Fraczyk and van Limbeek only rules out ε-expanders for any fixed ε > 0 and gives no rate; it does not imply exponential decay of the conductance. The paper is transparent about this dependence, but the abstract's phrase 'relate a general proof' should not be read as providing evidence for exponential thermalization in all non-amenable groups. Please make the conditional status of the general claim explicit in the abstract and in Sec. V B, and distinguish sharply between the proven hyperbolic-group case (if Theorem 4 is completed) and the conjectural general case.
minor comments (4)
  1. [Throughout] There are several typographical errors: 'resulting in in the slow diffusion' in Sec. I, 'Krlov graph' in the Fig. 2 caption and Sec. III A, 'minimial' in Sec. II B 2, 'exponetially' in Sec. VII 1, 'Fracyzk' for Fraczyk in Sec. V, and 'Kaimonovich' for Kaimanovich in App. C 2.
  2. [§IV A 1] The statement that diam(GK) = Θ(2^L) 'immediately implies' an exponentially long thermalization time uses a standard lower bound relating diameter to mixing time, but the bound is not stated or cited. Please add a one-sentence justification or a reference, since the implication is not immediate for graphs with large degree.
  3. [§IV C] The classification of the range-three dipole model as Class II is based on the heuristic that the real-space bottleneck yields exponentially slow thermalization, but no rigorous conductance bound is given. Please state explicitly that the class assignment is conjectural, or provide the missing bound.
  4. [App. B] The proof of the equivalence between exponential fragmentation and non-amenability (Propositions 7 and 8) relies on the technical assumption that a random word in K_e contains Θ(L) identity characters. This assumption is asserted to be 'very likely true' but is not proved; it should be isolated as a separate conjecture or lemma with a clear statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central conjecture is not equivalent to its inputs; conductance bounds come from explicit graph counting and independent mathematical theorems.

full rationale

The paper's load-bearing claim is that exponential fragmentation, defined by |Kmax|/|H| = O(exp(-L)) (Sec. II A), implies exponential thermalization time through the conductance bound tth >= C/Phi(GK) - 1 (Sec. III A, Eq. 11). This is a substantive conjectural link, not a definitional equivalence: none of the examples compute Phi(GK) from |Kmax|/|H| by construction. The tJz model obtains a binary-tree Krylov graph and computes the expansion of the cone C1 directly (Sec. IV B, App. E); the spin-1 breakdown model computes diam(GK) = Theta(2^L) directly from the charge sectors (Sec. IV A); the East model computes Phi(GK) = 1/(2(2N0-1)) by a lattice-path count (App. F). The hyperbolic-group argument invokes independent mathematical results (Fraczyk-van Limbeek, Bonk-Schramm, random-walk concentration, Kaimanovich), and while Theorem 4 is explicitly labeled informal and 'parts of our proof are a sketch' (Sec. V C), that is a rigor/completeness limitation, not a circular reduction. Self-citations to Refs. [24] and [33] are used for terminology, the pair-flip example, and the semigroup reformulation, but the central derivation does not reduce to those citations: the pair-flip result is independently reproduced in the framework, and the semigroup discussion extends, rather than assumes, the main conjecture. No fitted parameter is recycled as a prediction, and no uniqueness theorem by the authors is invoked to forbid alternatives. The paper's own caveats (Sec. II B, on the maximally-mixed fixed-point assumption) identify conditions under which the setup would need modification, which is an assumption about scope, not a circular step. Overall, the derivation chain is self-contained against explicit models and external mathematics, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted to data; thresholds such as gamma and epsilon are fixed constants in definitions. The main additional structure comes from mathematical theorems imported from group theory and expander graph theory, plus modeling assumptions about circuit-averaged dynamics. No new physical particles, forces, or conserved quantities are introduced; Krylov graphs and packed sectors are mathematical tools.

assumptions (8)
  • standard math Cheeger inequality for reversible Markov chains gives tth >= C(Phi^{-1} - 1).
    Used in Sec. III A, Eq. (11), to convert conductance upper bounds into thermalization time lower bounds.
  • domain assumption Uniform distribution is the stationary state of the circuit-averaged depolarizing dynamics.
    Assumed throughout Secs. II-III and used to define conductance; the paper notes this fails when Dyn has conserved quantities (Sec. II B).
  • domain assumption Instantaneous intrasector depolarization (Eq. 6) is the fastest bulk dynamics; local dynamics is no faster.
    Justifies focusing on the coarse-grained Krylov graph; the proof in App. A requires bistochastic local transition matrices.
  • standard math Gromov's theorem: random finitely presented groups are hyperbolic with high probability.
    Used in Sec. V A to argue that results for hyperbolic groups cover generic group-based constraints.
  • standard math Fraczyk-van Limbeek theorem: heat kernel measure is not epsilon-expanding on any bounded degree graph.
    Gives vanishing conductance but not an exponential rate; motivates Conjecture 2 in Sec. V B.
  • standard math Bonk-Schramm embedding and finite Assouad dimension of the Gromov boundary.
    Core geometric tools in App. C for proving exponential conductance bounds for hyperbolic groups.
  • ad hoc to paper Quantitative Benjamini conjecture (Conjecture 2).
    Introduced by the authors; proved for hyperbolic groups in Theorem 11, but not in general; the reduction from the main conjecture to this is exact for group and semigroup dynamics.
  • ad hoc to paper Technical assumption in App. B that words in K_e have O(L) identity characters and that filling length scaling holds as assumed.
    Used in Proposition 8 to rule out exponential fragmentation for amenable groups; authors call it very reasonable but leave a proof to future work.

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Pith. "Pith review of Exponentially slow thermalization in 1D fragmented dynamics." pith.science (2026). https://pith.science/paper/6DWPENPS

@misc{pith2026250113930,
  author       = {Pith},
  title        = {Pith review of: Exponentially slow thermalization in 1D fragmented dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DWPENPS}},
  note         = {Machine review of arXiv:2501.13930}
}
abstract

We investigate the thermalization dynamics of 1D systems with local constraints coupled to an infinite temperature bath at one boundary. The coupling to the bath eventually erases the effects of the constraints, causing the system to tend towards a maximally mixed state at long times. We show that for a large class of local constraints, the time at which thermalization occurs can be extremely long. In particular, we present evidence for the following conjecture: when the constrained dynamics displays strong Hilbert space fragmentation, the thermalization time diverges exponentially with system size. We show that this conjecture holds for a wide range of dynamical constraints, including dipole-conserving dynamics, the $tJ_z$ model, and a large class of group-based dynamics, and relate a general proof of our conjecture to a different conjecture about the existence of certain expander graphs.

Figures

Figures reproduced from arXiv: 2501.13930 by the authors.

Figure 1
Figure 1. FIG. 1. A schematic of the general setup. In [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Coarse-graining and the Krylov graph. Each vertex [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Different types of Hilbert space connectivity that [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Allowed dynamical moves of the spin-1 break [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Numerical results for the total charge relaxation dy [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Krylov graph of the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Magnetization relaxation time [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Numerical results for the range-three dipole [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Numerical results for the particle-conserving East [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Numerical results for the range-four dipole [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Computing the size of the thermalized Krylov sec [PITH_FULL_IMAGE:figures/full_fig_p040_13.png]

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