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arxiv: 2606.05293 · v1 · pith:KG4BYEPBnew · submitted 2026-06-03 · ❄️ cond-mat.str-el · cond-mat.dis-nn· quant-ph

Breakdown of Thermalization from Real-Time Dynamics in the Two-Dimensional Hubbard Model

Alessandro Sinibaldi , Luciano Loris Viteritti , Riccardo Rende , Fakher F. Assaad , Giuseppe Carleo This is my paper

Pith reviewed 2026-06-28 03:52 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.dis-nnquant-ph
keywords Hubbard modelthermalizationnonequilibrium dynamicsneural network quantum statesvariational Monte Carlodouble occupancyquench dynamicsstrongly correlated electrons
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The pith

Beyond a critical interaction strength, the two-dimensional Hubbard model fails to reach thermal equilibrium after an interaction quench.

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

The paper tracks the real-time evolution of double occupancy in the half-filled square-lattice Hubbard model after a sudden change in on-site repulsion. For weak to intermediate interactions the long-time value matches the prediction of a canonical thermal ensemble, while above a critical interaction the value deviates and thermalization signatures disappear. This distinction is obtained from time-dependent variational Monte Carlo simulations that employ transformer-based neural-network states to reach previously inaccessible long times. If correct, the result shows that strong correlations can block the usual relaxation to equilibrium in two-dimensional fermionic systems.

Core claim

In the half-filled two-dimensional Hubbard model, a quench in the interaction U produces two regimes: below a critical U_C the long-time double occupancy agrees with the thermal ensemble value, consistent with ergodic relaxation; above U_C the dynamics deviate from the thermal prediction and exhibit clear signatures of thermalization breaking.

What carries the argument

Time-dependent variational Monte Carlo combined with transformer-based Neural-Network Quantum States, which evolves the many-body wave function and extracts the long-time double occupancy without bias toward thermal or non-thermal states.

If this is right

  • Below the critical interaction the system relaxes to the thermal value of double occupancy.
  • Above the critical interaction the long-time double occupancy deviates from the thermal ensemble.
  • Numerical methods of this type can now access nonequilibrium regimes in correlated fermions that were previously unreachable.
  • The observed breakdown supplies a concrete diagnostic for thermalization failure in two-dimensional fermionic models.

Where Pith is reading between the lines

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

  • The critical interaction may mark the onset of a non-ergodic regime whose microscopic origin could be explored by varying lattice geometry or doping.
  • Quantum simulators could be tuned across the reported threshold to test whether the same deviation appears in measurable observables such as momentum distribution.
  • The result raises the question of whether similar thermalization breaking occurs after other quench protocols, such as changes in hopping or external fields.

Load-bearing premise

The variational Monte Carlo procedure with neural-network states yields an unbiased long-time limit for double occupancy that does not artificially favor either thermal or non-thermal behavior.

What would settle it

An ultracold-atom experiment measuring double occupancy at long times after an interaction quench in the 2D Hubbard model at half filling, performed both below and above the reported critical U value.

Figures

Figures reproduced from arXiv: 2606.05293 by Alessandro Sinibaldi, Fakher F. Assaad, Giuseppe Carleo, Luciano Loris Viteritti, Riccardo Rende.

Figure 1
Figure 1. Figure 1: Schematic illustration of the dynamical protocol studied to investigate thermalization. An initial Fermi sea state is [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Panel (a): Time evolution of the double occupancy ⟨dˆ⟩(t) [see Eq. (3)] for various interaction strengths U on a 8 × 8 lattice. The empty markers denote the tVMC simulations, while dashed lines indicate the dynamics extended via the time-dependent LVM. The shaded region highlights the time interval from which tVMC states are selected to perform the LVM computation. The arrows show the infinite-time limit o… view at source ↗
Figure 3
Figure 3. Figure 3: Effective temperature Teff as a function of the in￾teraction strength U for the dynamics with U = 1, 3, 4, 5 shown in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time evolution with tVMC (empty markers) of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution with tVMC of the double occupancy [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Double occupancy at infinite time ⟨dˆ⟩∞ for various interaction strengths U and system sizes 4 × 4, 6 × 6 and 8 × 8, computed using the time-dependent LVM, as a function of the final time tf of the tVMC simulation. Empty markers show the LVM data. The last data are fitted with the functional form a+b(e x/c−1) represented by solid lines, and filled markers indicate the extrapolated values in the limit tf → … view at source ↗
read the original abstract

Thermalization in strongly correlated fermionic systems remains a central open problem in quantum many-body physics. In this work, we investigate the real-time dynamics and the approach to thermalization in the two-dimensional Hubbard model, a paradigmatic framework for correlated electrons, relevant to high-temperature superconductivity and ultracold quantum simulation. Focusing on the half-filled square lattice, we monitor the time evolution of the double occupancy following a quench in the on-site interaction $U$, and assess whether its long-time value is captured by a canonical thermal ensemble. We employ time-dependent variational Monte Carlo methods combined with transformer-based Neural-Network Quantum States to accurately describe the nonequilibrium dynamics of fermions, especially for the behavior at long times, thereby accessing regimes that were previously inaccessible to numerical simulations. Our results reveal two distinct dynamical behaviors: for weak to intermediate interactions, the long-time double occupancy agrees with the thermal prediction, consistent with ergodic relaxation; beyond a critical interaction $U_{C}$, the dynamics deviate markedly from the thermal expectation, revealing clear signatures of thermalization breaking. These results establish numerical simulation as a powerful tool to probe nonequilibrium quantum phenomena in correlated fermionic matter.

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 / 1 minor

Summary. The paper studies real-time quench dynamics in the half-filled 2D Hubbard model on the square lattice using time-dependent variational Monte Carlo combined with transformer-based neural-network quantum states. It monitors the double occupancy and reports that its long-time value matches the prediction of a canonical thermal ensemble for weak-to-intermediate interaction strengths, while deviating markedly beyond a critical U_C, which the authors interpret as a breakdown of thermalization.

Significance. If the numerical evidence is robust, the result would constitute a concrete demonstration of non-ergodic long-time behavior in a paradigmatic fermionic lattice model, obtained from direct time evolution rather than equilibrium assumptions. The internal consistency check (recovery of thermal values in the ergodic regime) is a positive feature of the approach.

major comments (2)
  1. [Abstract / Methods] The abstract and available description supply no quantitative information on convergence with respect to variational parameters, bond dimension or network depth, finite-size scaling, or comparison against exactly solvable limits (e.g., U=0 or infinite-temperature states). Without these checks the claimed deviation for U>U_C cannot be assessed for systematic bias in the long-time limit.
  2. [Results] The identification of U_C and the statement that the dynamics 'deviate markedly' require explicit error bars, statistical uncertainties from the Monte Carlo sampling, and a clear criterion (e.g., a threshold on the difference from thermal value) that is independent of the variational ansatz.
minor comments (1)
  1. [Model and quench protocol] Notation for the quench protocol (initial state, sudden change in U) should be stated explicitly with a figure or equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the work's significance, and the constructive major comments. We address each point below and will revise the manuscript to strengthen the presentation of convergence, uncertainties, and criteria.

read point-by-point responses

  1. Referee: [Abstract / Methods] The abstract and available description supply no quantitative information on convergence with respect to variational parameters, bond dimension or network depth, finite-size scaling, or comparison against exactly solvable limits (e.g., U=0 or infinite-temperature states). Without these checks the claimed deviation for U>U_C cannot be assessed for systematic bias in the long-time limit.

    Authors: We agree that quantitative convergence information is needed to assess possible systematic bias. In the revised manuscript we will add a dedicated subsection with explicit data on convergence versus network depth and number of variational parameters. We will also include finite-size scaling for lattices up to 8x8 and direct comparisons to the exactly solvable U=0 limit (where long-time double occupancy recovers the thermal value) as well as the infinite-temperature state. These additions will allow readers to evaluate the robustness of the deviation reported for U>U_C. revision: yes

  2. Referee: [Results] The identification of U_C and the statement that the dynamics 'deviate markedly' require explicit error bars, statistical uncertainties from the Monte Carlo sampling, and a clear criterion (e.g., a threshold on the difference from thermal value) that is independent of the variational ansatz.

    Authors: We accept that error bars and an explicit, ansatz-independent criterion are required. The revised version will display statistical uncertainties from the Monte Carlo sampling on all relevant plots. We will define U_C via a concrete threshold (deviation from the thermal value exceeding three times the Monte Carlo error bar) and demonstrate that this threshold remains stable when the network architecture is varied. The criterion and its robustness will be stated clearly in the text and figures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical simulation result stands on direct computation

full rationale

The paper reports results from real-time t-VMC evolution with transformer NNQS on the 2D Hubbard model. The central claim (deviation from thermal double occupancy beyond U_C) is obtained by direct numerical time evolution and comparison to canonical ensemble values, not by any algebraic reduction, fitted-parameter renaming, or self-citation chain. The abstract explicitly notes internal consistency: agreement with thermal predictions at weak-to-intermediate U serves as a benchmark that the variational dynamics are unbiased. No equations or steps are presented that define a quantity in terms of itself or import uniqueness from prior self-work. The work is therefore self-contained against external thermal benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work rests on the standard Hubbard Hamiltonian and the variational ansatz but supplies no further ledger entries.

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