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arxiv: 2606.02713 · v1 · pith:RNILDYOAnew · submitted 2026-06-01 · ❄️ cond-mat.quant-gas · cond-mat.stat-mech· quant-ph

A mean-field description of strong-to-weak symmetry breaking in the monitored three-dimensional Bose-Hubbard model

Yicheng Tang , Pradip Kattel , J. H. Pixley This is my paper

Pith reviewed 2026-06-28 11:33 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas cond-mat.stat-mechquant-ph
keywords strong-to-weak symmetry breakingmonitored Bose-Hubbard modelcharge-sharpening transitionGutzwiller mean-fieldtrajectory-averaged order parameterquantum measurementsopen quantum systemsbosonic lattice
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The pith

A trajectory-averaged local order parameter detects strong-to-weak symmetry breaking in the monitored three-dimensional Bose-Hubbard model and becomes critical at the charge-sharpening transition.

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

The paper develops a Gutzwiller mean-field framework to simulate stochastic measurement dynamics directly in three-dimensional bosonic lattice systems. Applying it to the Bose-Hubbard model with local density measurements shows that strong-to-weak symmetry breaking can be tracked by a local order parameter averaged over quantum trajectories. This order parameter turns critical at the same measurement strength where charge sharpening occurs and displays a correlation-length exponent near 1.2 with Lorentz invariance. The shared location and scaling suggest the two phenomena share an underlying critical point, offering a local diagnostic in place of nonlocal ones.

Core claim

In the monitored three-dimensional Bose-Hubbard model, the trajectory-averaged local order parameter for strong-to-weak symmetry breaking becomes critical near the charge-sharpening transition, exhibits Lorentz invariance, and yields a correlation-length exponent ν ≃ 1.2 comparable to that of charge sharpening, indicating that the two may arise from a common critical point.

What carries the argument

Gutzwiller mean-field ansatz applied to stochastic quantum trajectories, with a local density-based order parameter averaged over measurement outcomes to identify strong-to-weak symmetry breaking.

If this is right

  • The local order parameter becomes critical at the same measurement strength as the charge-sharpening transition.
  • The order parameter displays Lorentz invariance and a correlation-length exponent of approximately 1.2.
  • The similarity in location and exponents points to a shared underlying critical point.
  • The mean-field method supplies concrete numerical predictions for experiments on monitored cold-atom systems.

Where Pith is reading between the lines

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

  • The same local-order-parameter construction could be tested in other monitored lattice models to check whether strong-to-weak symmetry breaking consistently coincides with charge sharpening.
  • If the two transitions are identical, monitored bosonic systems may possess only one relevant critical theory rather than separate ones.
  • Cold-atom experiments measuring both the local order parameter and charge fluctuations at varying measurement rates could directly test the predicted exponent.

Load-bearing premise

The Gutzwiller mean-field treatment remains quantitatively reliable for the stochastic measurement dynamics and correctly locates the transition in three dimensions.

What would settle it

Numerical or experimental data showing that the local order parameter stays non-critical at the measurement strength where charge sharpening occurs.

Figures

Figures reproduced from arXiv: 2606.02713 by J. H. Pixley, Pradip Kattel, Yicheng Tang.

Figure 1
Figure 1. Figure 1: FIG. 1: Illustration of the Bose-Hubbard model with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Panels (a)–(c) show the configuration of the superfluid order parameter Ψ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Configuration of the local superfluid order pa [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Upper four panels (a-d) show the time evolution for two order parameters, with varying measurement rate [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The left and middle panels show the time evolution of the R´enyi-2 correlator [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Data collapse of the dimensionless operators as a function of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

Strong-to-weak spontaneous symmetry breaking has emerged as a novel form of ordering in monitored and open quantum systems, yet its characterization has so far primarily relied on nonlocal diagnostics. Here, we develop a Gutzwiller mean-field framework for monitored bosonic lattice systems, enabling the direct simulation of stochastic measurement dynamics in three spatial dimensions. Applying this approach to the monitored Bose-Hubbard model with local density measurements and Lindbladian dissipation, we identify strong-to-weak symmetry breaking through a trajectory-averaged local order parameter. We find that this local order parameter becomes critical near the same measurement strength as the charge-sharpening transition and exhibits Lorentz invariance with a correlation-length exponent, $\nu\simeq 1.2$, comparable to that of the charge-sharpening transition, suggesting that the two phenomena may originate from a common underlying critical point. Our work establishes a local characterization of strong-to-weak symmetry breaking, reveals its connection to charge sharpening, and provides concrete predictions for future experiments on the monitored Bose-Hubbard model.

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 develops a Gutzwiller mean-field ansatz for the stochastic dynamics of the monitored 3D Bose-Hubbard model under local density measurements and Lindblad dissipation. It introduces a trajectory-averaged local order parameter to diagnose strong-to-weak spontaneous symmetry breaking, reports that this order parameter exhibits criticality at a measurement rate close to the charge-sharpening transition, and finds a correlation-length exponent ν ≃ 1.2 together with Lorentz invariance, suggesting the two transitions share an underlying critical point.

Significance. If the mean-field results are quantitatively reliable, the work supplies the first local diagnostic of strong-to-weak symmetry breaking and a concrete link to charge sharpening, together with experimentally testable predictions for the monitored Bose-Hubbard model in three dimensions. The extension of the Gutzwiller framework to stochastic trajectory evolution in 3D is a technical advance that enables direct simulation of the relevant dynamics.

major comments (2)
  1. [Methods / Results on the order-parameter criticality] The central claim that the strong-to-weak transition coincides with the charge-sharpening point and shares the exponent ν ≃ 1.2 rests on the accuracy of the site-factorized Gutzwiller ansatz for the stochastic Schrödinger equation. No benchmark against exact diagonalization, tensor-network methods, or fluctuation-corrected theories is provided to quantify how spatial correlations beyond mean field shift the critical measurement strength or renormalize the exponent in three dimensions near criticality.
  2. [Results section on the local order parameter] The reported value ν ≃ 1.2 and the statement of Lorentz invariance are obtained within the product-state ansatz; the manuscript does not report error bars, finite-size scaling collapse details, or a direct numerical comparison of the two transition locations that would substantiate the suggestion of a common underlying fixed point.
minor comments (1)
  1. [Abstract and §2] Notation for the trajectory-averaged order parameter and its scaling form should be defined explicitly with an equation number in the main text rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and the positive evaluation of our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Methods / Results on the order-parameter criticality] The central claim that the strong-to-weak transition coincides with the charge-sharpening point and shares the exponent ν ≃ 1.2 rests on the accuracy of the site-factorized Gutzwiller ansatz for the stochastic Schrödinger equation. No benchmark against exact diagonalization, tensor-network methods, or fluctuation-corrected theories is provided to quantify how spatial correlations beyond mean field shift the critical measurement strength or renormalize the exponent in three dimensions near criticality.

    Authors: We agree that the site-factorized Gutzwiller ansatz is a mean-field approximation whose quantitative accuracy near criticality may be affected by spatial fluctuations. In three dimensions, however, exact diagonalization and tensor-network methods remain computationally prohibitive for the stochastic trajectory dynamics considered here. The mean-field framework enables the first systematic study of this physics in 3D. We will add an explicit discussion of the limitations of the ansatz and the regime in which we expect it to remain qualitatively reliable. revision: partial

  2. Referee: [Results section on the local order parameter] The reported value ν ≃ 1.2 and the statement of Lorentz invariance are obtained within the product-state ansatz; the manuscript does not report error bars, finite-size scaling collapse details, or a direct numerical comparison of the two transition locations that would substantiate the suggestion of a common underlying fixed point.

    Authors: We will revise the results section to include the finite-size scaling collapse details, the procedure used to extract ν, and estimated uncertainties. We will also add a direct comparison (via plot or table) of the critical measurement strengths obtained from the local order parameter and from charge sharpening within the same Gutzwiller ansatz to better support the suggestion of a shared critical point. revision: yes

Circularity Check

0 steps flagged

No circularity: results are numerical outputs from independent mean-field dynamics

full rationale

The paper defines a Gutzwiller product-state ansatz, evolves it under an effective stochastic Schrödinger equation with local density measurements, and computes a trajectory-averaged local order parameter whose criticality is extracted numerically. The reported coincidence of the strong-to-weak transition with the charge-sharpening point and the value ν≃1.2 are direct simulation outputs, not quantities fitted or defined in terms of themselves. No load-bearing premise reduces to a self-citation chain, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via citation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the accuracy of the Gutzwiller ansatz for stochastic monitored dynamics; no explicit free parameters or new entities are named in the abstract.

axioms (1)
  • domain assumption Gutzwiller mean-field ansatz accurately captures the trajectory-averaged dynamics and critical behavior of the monitored 3D Bose-Hubbard model
    The entire simulation framework and the identification of the local order parameter rest on this approximation.

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