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arxiv: 2506.20617 · v2 · submitted 2025-06-25 · ❄️ cond-mat.other · physics.optics

Tunable lower critical fractal dimension for a non-equilibrium phase transition

Mattheus Burkhard , Luca Giacomelli , Cristiano Ciuti This is my paper

Pith reviewed 2026-05-19 08:22 UTC · model grok-4.3

classification ❄️ cond-mat.other physics.optics
keywords non-equilibrium phase transitionfractal dimensiondriven-dissipative bosonslower critical dimensionfrequency detuningHausdorff dimensioncritical slowing downmulti-mode dynamics
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The pith

The lower critical dimension of a non-equilibrium phase transition in a driven-dissipative bosonic system can be a non-integer fractal value continuously tuned by the driving frequency detuning.

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

This paper examines a driven-dissipative system of interacting bosons where the spatial dimension is set by the shape of the external driving field. Homogeneous drives give integer dimensions while fractal patterns produce non-integer Hausdorff dimensions for the density. Numerical simulations of the multi-mode dynamics show that the lower critical dimension, identified from how the asymptotic decay rate scales with system size, can itself be fractal. This dimension varies continuously as the frequency detuning of the drive is changed. The result is supported by a statistical mean-field analysis.

Core claim

The lower critical dimension of the non-equilibrium phase transition is shown to be tunable to non-integer fractal values by adjusting the frequency detuning of the driving field in a driven-dissipative bosonic system, where the spatial dimension is controlled by the driving field's shape, as determined from the system-size scaling of the asymptotic decay rate in multi-mode simulations and mean-field theory.

What carries the argument

The system-size dependence of the asymptotic decay rate extracted from numerical simulations of the full multi-mode dynamics, used to identify the lower critical dimension for both integer and fractal spatial densities.

Load-bearing premise

The assumption that the system-size dependence of the asymptotic decay rate extracted from numerical simulations of the full multi-mode dynamics reliably identifies the lower critical dimension even when the spatial density is fractal.

What would settle it

Numerical simulation of the decay rate scaling for a specific fractal driving pattern at varying detunings that fails to show continuous tuning of the lower critical dimension or deviates from the expected system-size behavior.

Figures

Figures reproduced from arXiv: 2506.20617 by Cristiano Ciuti, Luca Giacomelli, Mattheus Burkhard.

Figure 1
Figure 1. Figure 1: FIG. 1. Driving intensity patterns (top panels) and grayscale maps (bottom panels) of the corresponding steady-state bosonic [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Illustration of the approach to the thermodynamic [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Minimal asymptotic decay rate [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We theoretically investigate the role of spatial dimension and driving frequency in a non-equilibrium phase transition of a driven-dissipative interacting bosonic system. In this setting, spatial dimension is dictated by the shape of the external driving field. We consider both homogeneous driving configurations, which correspond to standard integer-dimensional systems, and fractal driving patterns, which give rise to a non-integer Hausdorff dimension for the spatial density. The onset of criticality is characterized by critical slowing down in the excited density dynamics as the system asymptotically approaches the steady state. By analyzing the system-size dependence of the asymptotic decay rate using numerical simulations of the full multi-mode dynamics, complemented by an analytical statistical mean-field treatment, we determine the lower critical dimension of the non-equilibrium phase transition. We show that this dimension can be non-integer and fractal in nature, and that it can be tuned continuously via the frequency detuning of the driving field.

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

1 major / 2 minor

Summary. The manuscript investigates non-equilibrium phase transitions in a driven-dissipative interacting bosonic system where the effective spatial dimension is set by the geometry of the external driving field. Both homogeneous (integer-dimensional) and fractal driving patterns (non-integer Hausdorff dimension) are considered. Criticality is diagnosed via critical slowing down in the excited-state density, and the lower critical dimension is extracted from the system-size scaling of the asymptotic decay rate obtained in numerical simulations of the full multi-mode dynamics, supplemented by a statistical mean-field treatment. The central result is that the lower critical dimension can take non-integer fractal values and can be tuned continuously by the frequency detuning of the drive.

Significance. If the finite-size scaling procedure remains valid when the spatial support is fractal, the work would provide a concrete example of a tunable, non-integer lower critical dimension in a non-equilibrium setting. The combination of direct multi-mode numerics with mean-field analysis is a positive feature that allows cross-validation of the scaling behavior.

major comments (1)
  1. [Numerical simulations and scaling analysis] Abstract and the paragraph describing the numerical procedure: the system-size dependence of the asymptotic decay rate is used to identify the lower critical dimension for both integer and fractal cases. For fractal driving fields the spatial density has Hausdorff dimension d_f < d, so the effective volume scales as L^{d_f} rather than L^d. Standard finite-size scaling forms for decay rates near criticality implicitly assume uniform Euclidean measure; it is not shown whether the fitting procedure or the functional form of the decay-rate scaling was re-derived or validated for non-integer Hausdorff measure. This assumption is load-bearing for the claim that the lower critical dimension itself can be fractal and continuously tuned.
minor comments (2)
  1. [Mean-field analysis] The mean-field treatment is described as 'statistical'; a brief derivation or explicit statement of the closure used would help readers assess its regime of validity relative to the multi-mode simulations.
  2. Figure captions for the decay-rate versus system-size plots should explicitly state the fitting window and the criterion used to declare that the decay rate has reached its asymptotic value.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. The major comment concerning the finite-size scaling procedure for fractal driving fields is addressed point by point below. We provide clarifications and commit to revisions where appropriate.

read point-by-point responses

  1. Referee: [Numerical simulations and scaling analysis] Abstract and the paragraph describing the numerical procedure: the system-size dependence of the asymptotic decay rate is used to identify the lower critical dimension for both integer and fractal cases. For fractal driving fields the spatial density has Hausdorff dimension d_f < d, so the effective volume scales as L^{d_f} rather than L^d. Standard finite-size scaling forms for decay rates near criticality implicitly assume uniform Euclidean measure; it is not shown whether the fitting procedure or the functional form of the decay-rate scaling was re-derived or validated for non-integer Hausdorff measure. This assumption is load-bearing for the claim that the lower critical dimension itself can be fractal and continuously tuned.

    Authors: We appreciate the referee highlighting this subtlety in applying finite-size scaling to non-Euclidean supports. In our simulations the system is discretized on a regular lattice of linear size L, with the driving field applied exclusively to the fractal subset whose support has Hausdorff dimension d_f. The asymptotic decay rate is extracted from the long-time relaxation of the excited-state density, averaged over the driven sites. We agree that a fully rigorous treatment requires adapting the scaling ansatz to the fractal measure rather than assuming uniform Euclidean volume. In the revised manuscript we will add a dedicated subsection that derives the generalized finite-size scaling form for the decay rate, incorporating the Hausdorff measure into the correlation volume and hyperscaling relations. This derivation shows that the leading L-dependence of the decay rate remains of the same functional form, with the effective dimension d_f entering the exponents. We benchmark the procedure on the integer-dimensional cases, where the extracted lower critical dimension reproduces known results for the model, and extend the same fitting protocol to the fractal patterns. The statistical mean-field treatment, being insensitive to microscopic spatial details, provides an independent consistency check. We will also include additional numerical evidence, such as attempted data collapse using d_f, to validate the robustness of the extracted lower critical dimension. revision: yes

Circularity Check

0 steps flagged

No circularity: lower critical dimension extracted from independent numerical scaling and mean-field analysis

full rationale

The paper determines the lower critical dimension via direct numerical extraction of the system-size dependence of the asymptotic decay rate in full multi-mode simulations, supplemented by a separate analytical statistical mean-field treatment. This chain does not reduce by construction to its own inputs: the fractal Hausdorff dimension arises from the external driving-field shape, the decay-rate scaling is computed from the dynamics rather than fitted to the target dimension, and no self-citation or ansatz is invoked as the load-bearing justification. The approach is presented as applicable to both integer and non-integer cases without tautological redefinition or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work relies on the validity of mean-field theory for the driven-dissipative bosonic system and on the numerical extraction of decay rates from finite-size simulations.

axioms (1)
  • domain assumption Mean-field statistical treatment captures the essential scaling of the asymptotic decay rate near criticality
    Abstract states the results are 'complemented by an analytical statistical mean-field treatment'

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Reference graph

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