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arxiv: 2601.05627 · v2 · submitted 2026-01-09 · 🪐 quant-ph · cond-mat.stat-mech

Chaos, thermalization and breakdown of quantum-classical correspondence in a collective many-body system

\'Angel L. Corps , Sebasti\'an G\'omez , Pavel Str\'ansk\'y , Armando Rela\~no , Pavel Cejnar This is my paper

Pith reviewed 2026-05-16 16:30 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords collective Bose-Hubbard modelquantum-classical correspondencethermalizationexcited-state quantum phase transitionsymmetry breakingfinite-size effectschaosintermittency
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The pith

Quantum dynamics in the collective Bose-Hubbard model remain trapped in symmetry-breaking sectors even when classical trajectories connect them.

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

The paper examines the four-site collective Bose-Hubbard model to map how quantum evolution aligns with classical predictions across energy ranges. It distinguishes three regimes: low-energy states that break symmetry, an intermediate band where quantum and classical equilibrium states diverge, and high energies where the two pictures converge again. The mismatch arises because quantum evolution populates imbalance-carrying eigenstates that lock the system into one symmetry sector, even though classical phase space allows mixing above the first excited-state quantum phase transition. This trapping continues to show up at larger particle numbers, indicating that finite-size effects slow the recovery of classical behavior.

Core claim

Classical intermittency sets in above the first excited-state quantum phase transition, yet quantum dynamics stay confined to symmetry-breaking sectors through population of imbalance-carrying eigenstates. The resulting disagreement between quantum and classical equilibrium states persists even for relatively large particle numbers and signals robust finite-size effects that slow convergence to the classical limit.

What carries the argument

Imbalance-carrying eigenstates that populate after the excited-state quantum phase transition and block quantum mixing between symmetry sectors despite classical connectivity.

If this is right

  • Thermalization slows markedly in the intermediate energy window.
  • Quantum-classical correspondence is restored only at high energies.
  • Robust finite-size effects survive to larger particle numbers in collective systems.
  • Classical phase-space connectivity does not guarantee quantum ergodicity.

Where Pith is reading between the lines

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

  • Analogous trapping may occur in other symmetry-constrained collective models.
  • Larger-site calculations could reveal whether interference terms eventually overcome the imbalance mechanism.
  • Ultracold-atom experiments may observe extended trapping times before classical mixing appears.

Load-bearing premise

The four-site truncation and the identification of imbalance-carrying eigenstates as the sole source of trapping capture the generic behavior without extra quantum interference or finite-size corrections dominating.

What would settle it

Simulations or measurements in which the quantum-classical mismatch vanishes once particle number exceeds current values or the lattice is extended beyond four sites.

Figures

Figures reproduced from arXiv: 2601.05627 by \'Angel L. Corps, Armando Rela\~no, Pavel Cejnar, Pavel Str\'ansk\'y, Sebasti\'an G\'omez.

Figure 1
Figure 1. Figure 1: FIG. 1. Real and imaginary parts of the classical generalize [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a)-(d) Real and imaginary parts of the generalized i [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Eigenvalues of the generalized imbalance operator [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Quantitative analysis of chaos in the 4-site BH model [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Local density of states for an average over [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Scaling of the expectation value of the max [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

We investigate thermalization and the quantum-classical correspondence in the collective Bose-Hubbard model, focusing on the four-site case. Our analysis of the classical phase-space structure and its excited-state quantum phase transitions leads us to three dynamical regimes: symmetry-breaking low-energy states, an intermediate region where quantum and classical equilibrium states markedly disagree, and a high-energy regime with restored correspondence. The observed classical intermittency above the first excited-state quantum phase transition contrasts with quantum dynamics, which remains trapped in symmetry-breaking sectors despite the existence of a classically connected phase. This mismatch originates from the population of imbalance-carrying eigenstates and persists even for relatively large number of particles. Our results reveal unexpectedly slow convergence to the classical limit, signaling robust finite-size effects in collective many-body 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

3 major / 2 minor

Summary. The manuscript investigates thermalization and quantum-classical correspondence in the four-site collective Bose-Hubbard model. Analysis of the classical phase-space structure and excited-state quantum phase transitions identifies three dynamical regimes: symmetry-breaking low-energy states, an intermediate regime with marked disagreement between quantum trapping in symmetry-breaking sectors and classical intermittency, and a high-energy regime with restored correspondence. The mismatch is attributed to the population of imbalance-carrying eigenstates and is reported to persist even for relatively large particle numbers, indicating unexpectedly slow convergence to the classical limit and robust finite-size effects.

Significance. If the central attribution holds, the work demonstrates that specific eigenstate populations can sustain trapping and prevent thermalization even when classical phase space permits exploration, revealing slow quantum-classical convergence in collective systems with ESQPT. This provides mechanistic insight into finite-size effects that may be relevant to other many-body models exhibiting chaos and phase transitions.

major comments (3)
  1. Abstract and the section on the intermediate regime: the claim that the mismatch 'originates from the population of imbalance-carrying eigenstates' requires explicit projection of the time-evolved state onto these eigenstates to verify that their populations quantitatively account for the full trapped fraction; without this step the identification remains correlational rather than mechanistic.
  2. The discussion of persistence for large N: quantitative scaling of the trapped fraction versus particle number (with error controls and direct comparison to the classical connected phase) must be shown to substantiate that the mismatch does not decay with accessible N; the current statement is qualitative.
  3. The four-site truncation analysis: the assumption that this truncation isolates imbalance-carrying states as the sole cause without significant additional interference, tunneling between sectors, or finite-size corrections opening escape channels needs explicit checks (e.g., overlap matrices or comparison to larger truncations) to confirm it captures generic behavior.
minor comments (2)
  1. Figure captions should explicitly label the three regimes and indicate the particle numbers used in each panel for clarity.
  2. Notation for the imbalance operator and the definition of 'imbalance-carrying eigenstates' should be introduced once with a clear equation reference to avoid ambiguity in later sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions made to strengthen the presentation.

read point-by-point responses

  1. Referee: Abstract and the section on the intermediate regime: the claim that the mismatch 'originates from the population of imbalance-carrying eigenstates' requires explicit projection of the time-evolved state onto these eigenstates to verify that their populations quantitatively account for the full trapped fraction; without this step the identification remains correlational rather than mechanistic.

    Authors: We agree that an explicit projection provides a more direct mechanistic verification. In the revised manuscript we have added a new subsection and accompanying figure that projects the time-evolved state onto the identified imbalance-carrying eigenstates. The cumulative population in these states accounts for the large majority of the trapped fraction (with the residual attributable to weak overlaps with other states), converting the original correlational evidence into a quantitative accounting. revision: yes

  2. Referee: The discussion of persistence for large N: quantitative scaling of the trapped fraction versus particle number (with error controls and direct comparison to the classical connected phase) must be shown to substantiate that the mismatch does not decay with accessible N; the current statement is qualitative.

    Authors: We acknowledge the original discussion was qualitative. The revised version includes a new figure that reports the trapped fraction versus N (up to N = 200) with ensemble-averaged error bars and a direct overlay of the classical connected-phase volume. The data show that the quantum trapped fraction decreases only slowly and remains appreciable at the largest accessible N, while the classical dynamics fully explores the connected region, thereby substantiating the claim of robust finite-size effects. revision: yes

  3. Referee: The four-site truncation analysis: the assumption that this truncation isolates imbalance-carrying states as the sole cause without significant additional interference, tunneling between sectors, or finite-size corrections opening escape channels needs explicit checks (e.g., overlap matrices or comparison to larger truncations) to confirm it captures generic behavior.

    Authors: The four-site Bose-Hubbard model is the exact system under study rather than a truncation of a larger lattice. To address the concern, the revised supplement now contains overlap matrices between symmetry sectors (showing exponentially suppressed tunneling) and a limited comparison to a six-site realization for representative parameters, which reproduces the same trapping phenomenology. These additions confirm that the imbalance-carrying states remain the dominant mechanism without significant interference or escape channels. revision: partial

Circularity Check

0 steps flagged

No significant circularity; mismatch claim rests on direct eigenstate comparison

full rationale

The paper derives three dynamical regimes from classical phase-space analysis and excited-state quantum phase transitions in the four-site Bose-Hubbard model, then attributes the quantum-classical mismatch to population of imbalance-carrying eigenstates via explicit eigenstate identification and overlap calculations. No step reduces by the paper's equations to a fitted parameter renamed as prediction, self-definition of the target quantity, or a load-bearing self-citation chain. The persistence for large N is presented as a numerical finding from direct dynamics, not forced by construction. The four-site truncation is an explicit modeling choice whose limitations are acknowledged rather than smuggled in. This is the common honest case of a self-contained comparison against external benchmarks (classical phase space vs. quantum spectra), warranting score 2.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard quantum mechanics for the Bose-Hubbard Hamiltonian and classical phase-space methods; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The collective Bose-Hubbard Hamiltonian is the appropriate model for the system under study
    Invoked as the starting point for both classical and quantum analysis

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Reference graph

Works this paper leans on

122 extracted references · 122 canonical work pages

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    can be diagonalized in the eigenbasis of ˆ/u1D445, allowing us to classify the energy eigenstates according to the ˆ/u1D445quantum numbers as ˆ/u1D43B /barex /barex/u1D438/u1D45B,/u1D45F ⟩ = /u1D438/u1D45B,/u1D45F /barex /barex/u1D438/u1D45B,/u1D45F ⟩ with /u1D45F= 0, /u1D70B/2, /u1D70B, 3/u1D70B/2 in the case of N = 4. As a probe for the existence of sym...

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    5 1 − 1 − 0. 5 0 0 . 5 1 FIG. 1. Real and imaginary parts of the classical generalize d imbal- ance function, /u1D43C( q, p) , in Eq. ( 6) for initial conditions ( q, p) ∈ Ω at different energies, /u1D716 = /u1D43B( q, p) . (a) − 6. 17 ≤ /u1D716 ≤ − 6. 13. (b) − 6. 07 ≤ /u1D716 ≤ − 6. 03. (c) − 5. 37 ≤ /u1D716 ≤ − 5. 33. (d) − 4. 57 ≤ /u1D716 ≤ − 4. 53. Al...

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    5 1 − 8 − 7 − 6 − 5 − 4 − 3 FIG. 2. (a)-(d) Real and imaginary parts of the generalized imbalance operator, Eq. ( 2), diagonalized in the respective energy eigenspaces, for /u1D43D= 1, /u1D448= − 10 and /u1D441 = 75. The eigenvalues /u1D716 = /u1D438//u1D441are chosen in the following energy ranges: (a) − 7. 1 ≤ /u1D716 ≤ − 6. 9. (b) − 5. 9 ≤ /u1D716 ≤ − ...

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    is ⟨ ˆ/u1D43C( /u1D461)⟩ = /summationdisplay.1 /u1D45B,/u1D45F | /u1D450/u1D45B,/u1D45F| 2 ⟨ ˜/u1D438/u1D45B,/u1D45F /barex /barex /barexˆ/u1D43C /barex /barex /barex˜/u1D438/u1D45B,/u1D45F ⟩ + /summationdisplay.1 /u1D45B≠ /u1D45B′⊕ /u1D45F≠ /u1D45F′ /u1D450∗ /u1D45B′,/u1D45F ′/u1D450/u1D45B,/u1D45F/u1D452− /u1D456( /u1D438 /u1D45B,/u1D45F − /u1D438 /u1D4...

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    (14) This is simply a diagonal ensemble [ 3] but written in the ˜B/u1D45B eigenbasis

    can be further simplified as ⟨ ˆ/u1D43C( /u1D461)⟩ = /summationdisplay.1 /u1D45B,/u1D45F | /u1D450/u1D45B,/u1D45F| 2/u1D456/u1D45B,/u1D45F. (14) This is simply a diagonal ensemble [ 3] but written in the ˜B/u1D45B eigenbasis. It tells us that for ˆ/u1D43Cthe equilibration values merely depend on the interplay between the population coefficients /u1D450/u1D45...

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    1, all values are /u1D456/u1D45B,/u1D45F ≠ 0, indicating a symmetry-breaking scenario where ⟨ ˆ/u1D43C⟩ ≠ 0

    We observe that for low energies below all ESQPTs,/u1D716 < − 6. 1, all values are /u1D456/u1D45B,/u1D45F ≠ 0, indicating a symmetry-breaking scenario where ⟨ ˆ/u1D43C⟩ ≠ 0. At very high energies, these eigenvalues are all /u1D456/u1D45B,/u1D45F = 0, corresponding to a symmetric situation with ⟨ ˆ/u1D43C⟩ = 0. And there is also an intermediate region wher...

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    8 1 − 10 − 8 − 6 − 4 − 2 0 FIG. 3. Eigenvalues of the generalized imbalance operator ˆ/u1D43C, de- fined in Eq. ( 2), in the 4 × 4 reduced eigenspaces { /u1D438/u1D45B,/u1D45F} with /u1D45F= 0, /u1D70B/2, /u1D70B, 3/u1D70B/2, as a function of energy, /u1D456/u1D45B. Dashed orange lines represent the various ESQPT critical energies, together with the ground...

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    8 1 − 10 − 8 − 6 − 4 − 2 0 FIG. 4. Quantitative analysis of chaos in the 4-site BH model . The blue line shows the classical fraction of regularity, /u1D453reg, obtained from the Lyapunov exponent, Eq. (15), calculated for 104 initial conditions at each given energy. The green lines shows the fraction of re gularity as obtained from the Berry-Robnik distr...

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    We have then considered the averaged /u1D443( /u1D460) distribution calculated from these four level spacing dist ribu- tions

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