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arxiv: 2401.11294 · v2 · submitted 2024-01-20 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

Exponentially slow thermalization and the robustness of Hilbert space fragmentation

Yiqiu Han , Xiao Chen , Ethan Lake This is my paper

Pith reviewed 2026-05-24 04:40 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el
keywords Hilbert space fragmentationslow thermalizationrandom unitary circuitsconfiguration space bottlenecksboundary bathpair-flip constraintentanglement dynamics1d quantum chains
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The pith

A boundary bath leaves a pair-flip constrained quantum chain unthermalized for a time that grows exponentially with system size.

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

The paper studies a one-dimensional chain whose pair-flip constraint fragments Hilbert space into disconnected sectors, then couples one end to an ergodicity-restoring thermal bath. Hamiltonian simulations show that both entanglement growth and the relaxation of local observables remain exponentially slow in system size. To explain the scaling, the authors map the problem onto an exactly solvable random unitary circuit model with the same constraint. In that circuit they prove that thermalization time scales exponentially with length because the allowed moves create strong bottlenecks that keep most configurations unreachable for exponentially long times. This supplies a concrete dynamical mechanism by which approximate fragmentation can produce anomalously slow relaxation even when perfect isolation is absent.

Core claim

In the random unitary circuit model of a pair-flip constrained chain with boundary bath coupling, thermalization time scales exponentially with system size because strong bottlenecks in configuration space restrict how the system can explore Hilbert space.

What carries the argument

Strong bottlenecks in configuration space within the random unitary circuit model with pair-flip constraints and boundary coupling.

If this is right

  • Entanglement entropy grows only after an exponentially long waiting time.
  • Local observables remain far from their thermal values for exponentially long times.
  • The same exponential bottleneck persists when the bath is attached at the boundary rather than distributed throughout the chain.
  • Slow thermalization appears even though the bath is designed to restore ergodicity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Approximate versions of fragmentation may produce long-lived non-thermal states in open quantum systems of practical size.
  • Similar configuration-space bottlenecks could appear in other locally constrained models once a weak bath is added.
  • The mechanism offers a route to engineer slow relaxation without exact conservation laws.
  • Quantum simulators with tunable pair-flip rules could directly measure the predicted exponential timescale.

Load-bearing premise

The random unitary circuit dynamics with the chosen pair-flip constraint and boundary bath coupling faithfully captures the essential slow-relaxation mechanism of the underlying Hamiltonian system.

What would settle it

A direct simulation of the random unitary circuit model that finds thermalization time scaling polynomially rather than exponentially with system size for accessible lengths would falsify the central claim.

Figures

Figures reproduced from arXiv: 2401.11294 by Ethan Lake, Xiao Chen, Yiqiu Han.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: c. This suggests that the “outward” bias—which acts to slow thermalization—wins out. To understand why, we shift our focus from Hamiltonian to RU dynam￾ics, where rigorous bounds on thermalization times can be proven. RU dynamics and Hilbert space bottlenecks: To sim￾plify the dynamics, we replace the unconstrained region by a thermal bath which subjects the spin on the end of the chain to depolarizing noi… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The charge relaxation time [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical results of the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Eigenenergies in different variants of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Eigenenergies in different [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Eigenenergies for [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Scaling of the size of the largest Krylov sector [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The asymptotic expression ( [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The krylov graph for [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The number of states in the cone [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Markov gaps for [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The quantum numbers of each Krylov sector under the symmetries [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Markov gaps for [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
read the original abstract

The phenomenon of Hilbert space fragmentation, whereby dynamical constraints fragment Hilbert space into many disconnected sectors, provides a simple mechanism by which thermalization can be arrested. However, little is known about how thermalization occurs in situations where the constraints are not exact. To study this, we consider a situation in which a fragmented 1d chain with pair-flip constraints is coupled to an ergodicity-restoring thermal bath at its boundary. We numerically observe an exponentially long thermalization time in Hamiltonian dynamics, manifested in both entanglement dynamics and the relaxation of local observables. To understand this, we study an analogous model of random unitary circuit dynamics, whose thermalization time we prove scales exponentially with system size. Slow thermalization in this model is shown to be a consequence of strong bottlenecks in configuration space, which restrict how the system can explore Hilbert space, and demonstrate a new way of producing anomalously slow thermalization dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper examines a 1D chain with pair-flip constraints coupled to a boundary thermal bath. It reports numerical observations of exponentially long thermalization times in the Hamiltonian dynamics, visible in entanglement growth and local observable relaxation. An analogous random unitary circuit model is analyzed, for which the authors prove that thermalization time scales exponentially with system size due to strong bottlenecks in configuration space that limit Hilbert-space exploration.

Significance. If the results hold, the work supplies both numerical evidence and a rigorous proof of exponentially slow thermalization arising from approximate Hilbert-space fragmentation. The proof for the circuit model, grounded in configuration-space bottlenecks rather than fitted parameters, is a clear strength and illustrates a distinct mechanism for anomalously slow dynamics. This could inform studies of prethermalization in constrained many-body systems, provided the circuit faithfully reproduces the Hamiltonian bottlenecks.

major comments (2)
  1. [abstract (circuit-model paragraph) and Hamiltonian-numerics section] The exponential scaling is proven only for the random unitary circuit model; the Hamiltonian claim rests on numerical observation. The manuscript invokes the circuit to interpret the Hamiltonian results (abstract, circuit-model paragraph), but the load-bearing assumption that the pair-flip constraint plus boundary coupling produces equivalent bottlenecks in both models is not accompanied by a direct comparison of configuration-space exploration or relaxation pathways between the discrete-time circuit and continuous-time Hamiltonian.
  2. [numerical results for Hamiltonian dynamics] The numerical evidence for exponential scaling in the Hamiltonian dynamics should include explicit details on the range of system sizes, the functional form fitted to the relaxation times, and quantitative error analysis to distinguish exponential from sub-exponential behavior (e.g., Table or Figure reporting the scaling).
minor comments (2)
  1. Figure captions should explicitly state the system sizes and fitting procedures used to extract the exponential scaling.
  2. Notation for the pair-flip constraint and bath coupling should be made identical between the Hamiltonian and circuit sections to facilitate direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [abstract (circuit-model paragraph) and Hamiltonian-numerics section] The exponential scaling is proven only for the random unitary circuit model; the Hamiltonian claim rests on numerical observation. The manuscript invokes the circuit to interpret the Hamiltonian results (abstract, circuit-model paragraph), but the load-bearing assumption that the pair-flip constraint plus boundary coupling produces equivalent bottlenecks in both models is not accompanied by a direct comparison of configuration-space exploration or relaxation pathways between the discrete-time circuit and continuous-time Hamiltonian.

    Authors: We agree that the rigorous proof of exponential scaling applies exclusively to the random unitary circuit model, while the Hamiltonian results rely on numerical observations. The circuit is introduced as an analogous discrete-time model to provide mechanistic insight into the bottlenecks observed numerically in the continuous-time Hamiltonian dynamics. We acknowledge that the manuscript does not include a direct side-by-side comparison of configuration-space exploration or relaxation pathways. In the revised manuscript we will add such a comparison (e.g., by tracking the fraction of accessible configurations or effective diffusion rates in both models) to better substantiate the analogy. revision: yes

  2. Referee: [numerical results for Hamiltonian dynamics] The numerical evidence for exponential scaling in the Hamiltonian dynamics should include explicit details on the range of system sizes, the functional form fitted to the relaxation times, and quantitative error analysis to distinguish exponential from sub-exponential behavior (e.g., Table or Figure reporting the scaling).

    Authors: We thank the referee for this suggestion. The current manuscript presents numerical data on entanglement growth and local-observable relaxation, but the details of system-size range, explicit functional fits, and quantitative error analysis are not presented in a consolidated form. In the revision we will add a dedicated table (or supplementary figure) that lists the system sizes examined, the fitted forms (e.g., relaxation time versus system size), and statistical measures such as fit uncertainties or bootstrap errors to allow readers to assess the exponential versus sub-exponential distinction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central result is a rigorous proof of exponential thermalization time in the random unitary circuit model, derived from analysis of configuration-space bottlenecks rather than any fitted parameters, self-referential definitions, or self-citations. The Hamiltonian dynamics are addressed via separate numerical observations, with the circuit presented as an analogous model under an explicit modeling assumption. No load-bearing step reduces by construction to the paper's inputs, and the derivation does not rely on prior author work for uniqueness or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of the pair-flip constraint, the boundary bath coupling, and standard properties of random unitary circuits used to model the Markov process on configurations. No free parameters are introduced to fit the exponential scaling; no new entities are postulated.

axioms (1)
  • standard math Random unitary circuits with the stated pair-flip constraint generate a Markov chain on configuration space whose mixing time can be bounded by bottleneck analysis.
    Invoked to prove the exponential scaling of thermalization time.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum Fragmentation

    quant-ph 2026-04 unverdicted novelty 7.0

    A Rokhsar-Kivelson-type construction turns classical or non-fragmented models into quantum fragmented Hamiltonians whose Krylov sectors require an entangled basis to resolve.

Reference graph

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    Entanglement and Krylov entropy In this subsection we show the entanglement dynamics of a more general locally-perturbed PF model, including the half-chain entanglement entropy of the constrained system, and Krylov entropy, a quantity that measures the spreading of the wave function over different Krylov subspaces, i.e. SK(t) ≡ − X vs pvs ln pvs , (B2) wh...

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    r statistics The models with N > 2 that we study all exhibit strong HSF, with |Kmax| being exponentially smaller than |H|. This implies that if one examines the spectrum of H, consecutive eigenstates will almost certainly belong to distinct Krylov sectors, and hence the spectrum of H will exhibit no nearest-neighbor level repulsion in the absence of const...

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    N = 2 We first dispatch with the easy case of N = 2, for which the dynamics is not fragmented. The tree T2 is simply a line, and the different Krylov sectors can be fully distinguished by the charge Q1 defined in (2), the value of which gives the distance of the Krylov sector along the line. The number of Kyrlov sectors is simply NK(L) = L + 1. (C1) The d...

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    Pair-flip For general pair-flip dynamics, the elementary gates Ui,i+1 take the form Ui,i+1 = X a,b U P F ab |aa⟩⟨bb| + X a̸=b eiϕab |ab⟩⟨ab| = U P F ⊕ M a̸=b eiϕab , (D3) where the matrix U P F is drawn from the Haar ensemble on U(N), and the second term in the direct sum acts as a diagonal matrix of random phases on the subspace of states frozen under th...

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    and the slightly modified treatment given in [35]. We let O(t) ≡ E Ct [Ct(O)] (D4) denote the circuit-averaged evolution of O over time t, where the ECt denotes averaging over the unitaries constituting U. To help with notation, we will divide each unit time interval into three steps of length t = 1 /3: at t ∈ N the depolarizing noise is applied, at t ∈ N...

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    product states

    Temperley-Lieb For Temperley-Lieb dynamics, the elementary unitary gates are constrained to preserve the pair-flip constraint in any onsite basis, which is done by enriching the previously studied pair-flip dynamics with SU (N) symmetry. The elementary unitary gates for this model take the form Ui,i+1 = eiϕi,i+1ΠT L + (1 − ΠT L), (D12) where ΠT L ≡ 1 N X ...

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    A key notion in what follows will be that of the expansion: Definition 1

    Markov chains and graph expansion We begin by reviewing some central concepts in the theory of Markov chains, 2 letting M to denote the generator of a given Markov process. A key notion in what follows will be that of the expansion: Definition 1. Let R ⊂ H . The expansion of R is defined as the amount of probability flow that the uniform distribution expe...

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    non-local

    Local and non-local chains The Markov generator derived in Sec. D was obtained by considering brickwork RU dynamics composed of three alternating layers: one layer of constrained 2-site gates on the even sublattice ( Me in the notation of Sec. D), one layer on the odd sublattice ( Mo), and one layer consisting solely of depolarizing noise applied to the s...

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    The case of N = 2, where thermalization is expected to be much faster, is dealt with in a subsequent subsection

    N > 2 In this subsection we prove a variety of results establishing exponentially slow thermalization in the strongly fragmented models obtained when N > 2. The case of N = 2, where thermalization is expected to be much faster, is dealt with in a subsequent subsection. a. The spectral gap We begin by proving the following theorem: Theorem 4. For N > 2, th...

    work page

  56. [56]

    between” two Krylov sectors, rather than cutting “within

    N = 2 We now compute the expansion for N = 2. In this case we expect a large expansion—and hence a fast mixing time—due to the absence of strong Hilbert space bottlenecks. In this subsection we will find it most convenient to label the Krylov sectors by their charge Q, defined as Q ≡ X i (−1)iZi, (E75) with ⟨Q⟩ψ measuring the endpoint of the random walk d...

    work page