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arxiv: 2410.01892 · v2 · pith:2VJT5URRnew · submitted 2024-10-02 · ✦ hep-ph · cond-mat.quant-gas· hep-th

Superfluids in expanding backgrounds and attractor times

Guri K. Buza , Toshali Mitra , Alexander Soloviev This is my paper

Pith reviewed 2026-05-23 20:02 UTC · model grok-4.3

classification ✦ hep-ph cond-mat.quant-gashep-th
keywords superfluidhydrodynamic attractorsexpanding backgroundsBjorken flowGubser flowspontaneous symmetry breakingattractor timeMüller-Israel-Stewart theory
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The pith

In expanding backgrounds, hydrodynamic evolution of a superfluid can spontaneously break U(1) symmetry and then follow attractor dynamics for a finite attractor time set by initial conditions.

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

The paper shows that in a U(1) superfluid coupled to Müller-Israel-Stewart hydrodynamics, the fluid variables in expanding flows can reshape the Goldstone mode potential. This triggers spontaneous symmetry breaking for suitable starting data, after which the condensate shrinks and the system tracks hydrodynamic attractors. The authors introduce an attractor time that marks how long this regime lasts in Bjorken and Gubser expansions, and they give the first full nonlinear account of Gubser flow including a constant-anisotropy phase. Similar attractor behavior appears in a dynamical FLRW background, where the system later becomes condensate-dominated.

Core claim

For suitable initial conditions, the evolution of the hydrodynamic variables leads to a change in the potential of the Goldstone mode, spontaneously breaking the symmetry. After some time, the condensate becomes small, leading the system evolution to be well described via hydrodynamic attractors for a timescale that we determine in Bjorken and Gubser flows. We define this new timescale as the attractor time and show its dependence on initial conditions. In the case of the Gubser flow, we provide for the first time a complete description of the nonlinear evolution of the system, including a novel nonlinear regime of constant anisotropy not found in the Bjorken evolution.

What carries the argument

The attractor time: the interval after condensate suppression during which the coupled Goldstone-hydrodynamic system in Müller-Israel-Stewart theory follows attractor solutions in expanding geometries.

If this is right

  • The attractor time is finite and explicitly depends on the choice of initial hydrodynamic and Goldstone data.
  • Gubser flow admits a previously unseen nonlinear regime of constant anisotropy that persists while the system is on the attractor.
  • In FLRW expansion the same initial-condition-driven attractor is reached, but the late-time state is dominated by the condensate rather than hydrodynamics.
  • The mechanism supplies a concrete way to connect out-of-equilibrium superfluid dynamics to standard hydrodynamic attractors in heavy-ion and cosmological settings.

Where Pith is reading between the lines

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

  • The same initial-condition trigger could be tested in other expanding geometries or with different equation-of-state choices to map how attractor time scales with expansion rate.
  • If realized in a heavy-ion context, the transient symmetry-breaking window might leave observable imprints on particle spectra before the attractor regime takes over.
  • The constant-anisotropy phase unique to Gubser flow suggests that global flow geometry can qualitatively alter the nonlinear attractor structure beyond what Bjorken flow captures.

Load-bearing premise

Suitable initial conditions exist such that hydrodynamic evolution alters the Goldstone potential enough to break symmetry and suppress the condensate before other processes intervene.

What would settle it

A numerical evolution starting from the claimed initial data in which the effective potential never drives spontaneous symmetry breaking or the condensate remains large enough to block the attractor description throughout the expansion.

Figures

Figures reproduced from arXiv: 2410.01892 by Alexander Soloviev, Guri K. Buza, Toshali Mitra.

Figure 1
Figure 1. Figure 1: FIG. 1. The condensate (red point) in the potential ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Overview showing the relationships between the different backgrounds explored here. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Left: Typical evolution for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The normalized timescale to go from the onset of hydrodynamic attractor behavior, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The figure shows temperature [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The evolution of the anisotropy, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Left: Evolution of the anisotropy, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The figure shows the evolution of condensate [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The behavior of the attractor time, ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Behavior of ∆ [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The top left and right figure show the ∆ [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Mapping the Gubser solution in Fig. [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The dissipative fluid in FLRW with [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The evolution of condensate [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The evolution of the full system in the FLRW background. The top two plots show the evolution of the full system [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Phase space plot in the FLRW background. The initial conditions are such that [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Left: The decaying amplitude [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
read the original abstract

We determine the behavior of an out-of-equilibrium superfluid, composed of a $U(1)$ Goldstone mode coupled to hydrodynamic modes in a M\" uller-Israel-Stewart theory, in expanding backgrounds relevant to heavy ion collision experiments and cosmology. For suitable initial conditions, the evolution of the hydrodynamic variables leads to a change in the potential of the Goldstone mode, spontaneously breaking the symmetry. After some time, the condensate becomes small, leading the system evolution to be well described via hydrodynamic attractors for a timescale that we determine in Bjorken and Gubser flows. We define this new timescale as the \textit{attractor time} and show its dependence on initial conditions. In the case of the Gubser flow, we provide for the first time a complete description of the nonlinear evolution of the system, including a novel nonlinear regime of constant anisotropy not found in the Bjorken evolution. Finally, we consider the superfluid in the dynamical FLRW (Friedmann-Lemaitre-Roberston-Walker) background, where we observe a similar attractor behavior, dependent on the initial conditions, that at late times approaches a regime dominated by the condensate.

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 manuscript examines the coupled dynamics of a U(1) Goldstone mode with Müller-Israel-Stewart hydrodynamics for an out-of-equilibrium superfluid in expanding backgrounds (Bjorken, Gubser, and dynamical FLRW). It asserts that, for suitable initial conditions, the hydrodynamic evolution modifies the effective potential of the Goldstone mode, inducing spontaneous symmetry breaking; the condensate subsequently suppresses, allowing the system to enter a hydrodynamic attractor regime over a newly defined 'attractor time' whose dependence on initial data is reported. For Gubser flow the work supplies a complete nonlinear evolution, including a novel regime of constant anisotropy absent in Bjorken flow; a similar attractor behavior is observed in FLRW, with late-time condensate domination.

Significance. If the numerical results are robust, the identification of an attractor time and the first complete nonlinear treatment of Gubser flow with a constant-anisotropy regime constitute concrete advances for modeling superfluids in heavy-ion and cosmological settings. The explicit coupling of Goldstone dynamics to MIS hydrodynamics in expanding geometries is a timely extension, and the provision of attractor timescales dependent on initial conditions supplies falsifiable predictions once the initial-data basin is quantified.

major comments (2)
  1. [Abstract / Sec. 1] The central claim that hydrodynamic evolution drives a change in the Goldstone potential sufficient to trigger spontaneous symmetry breaking (and subsequent condensate suppression) is stated to hold only for 'suitable initial conditions' (abstract and introduction). Because the attractor time is defined from the duration of this regime and is reported to depend on those initial data, the manuscript must demonstrate the size of the basin of attraction or provide a parameter scan showing that the potential modification is not an artifact of specially chosen starting values; without this the definition of the attractor time remains conditional on an unquantified assumption.
  2. [Gubser-flow section] The novel constant-anisotropy nonlinear regime reported for Gubser flow is load-bearing for the claim of a 'complete description' not found in Bjorken evolution. The manuscript should supply the explicit evolution equations or numerical diagnostics (e.g., the anisotropy parameter as a function of proper time) that establish this regime is reached from the coupled Goldstone-hydro system rather than imposed by hand, together with error estimates on the attractor time extracted from it.
minor comments (2)
  1. [Abstract] The abstract would benefit from a single sentence indicating the numerical method (e.g., characteristic or relaxation scheme) and the range of initial hydrodynamic and Goldstone variables explored.
  2. [Sec. 2] Notation for the effective potential of the Goldstone mode and its coupling to the hydrodynamic stress tensor should be introduced with an equation reference at first use to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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

read point-by-point responses

  1. Referee: [Abstract / Sec. 1] The central claim that hydrodynamic evolution drives a change in the Goldstone potential sufficient to trigger spontaneous symmetry breaking (and subsequent condensate suppression) is stated to hold only for 'suitable initial conditions' (abstract and introduction). Because the attractor time is defined from the duration of this regime and is reported to depend on those initial data, the manuscript must demonstrate the size of the basin of attraction or provide a parameter scan showing that the potential modification is not an artifact of specially chosen starting values; without this the definition of the attractor time remains conditional on an unquantified assumption.

    Authors: We agree that the robustness of the attractor time would be strengthened by a more systematic quantification of the basin of attraction. The manuscript already illustrates the phenomenon and the dependence of the attractor time on initial data for several representative choices that trigger the symmetry-breaking regime. To address the referee's concern directly, we will add an expanded parameter scan over initial conditions in the revised version, delineating the region in which the hydrodynamic modification of the Goldstone potential leads to spontaneous symmetry breaking. revision: yes

  2. Referee: [Gubser-flow section] The novel constant-anisotropy nonlinear regime reported for Gubser flow is load-bearing for the claim of a 'complete description' not found in Bjorken evolution. The manuscript should supply the explicit evolution equations or numerical diagnostics (e.g., the anisotropy parameter as a function of proper time) that establish this regime is reached from the coupled Goldstone-hydro system rather than imposed by hand, together with error estimates on the attractor time extracted from it.

    Authors: The evolution equations for the Gubser-flow case are obtained by coupling the Goldstone mode to the MIS hydrodynamic equations in the appropriate coordinates and are solved numerically in Section 3. We will include the explicit set of coupled ODEs in an appendix of the revised manuscript. In addition, we will provide plots of the anisotropy parameter as a function of proper time together with numerical convergence diagnostics that confirm the constant-anisotropy regime emerges dynamically from the coupled system; these will also include error estimates on the extracted attractor times. revision: yes

Circularity Check

0 steps flagged

No circularity: forward evolution from initial conditions defines attractor time without reduction to inputs

full rationale

The paper evolves the coupled Müller-Israel-Stewart hydrodynamics plus Goldstone mode from specified initial conditions in Bjorken, Gubser and FLRW backgrounds. The attractor time is introduced as the duration after which the condensate amplitude drops and the evolution enters the hydrodynamic attractor regime; this is an observed outcome of the time integration rather than a quantity fitted to the same data or defined in terms of itself. No self-citations, uniqueness theorems, or ansätze are invoked to force the result, and the novel constant-anisotropy regime in Gubser flow is reported as a direct numerical feature. The assumption of 'suitable initial conditions' is an existence claim about the basin of attraction, not a tautological redefinition of the output quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard assumptions of Müller-Israel-Stewart hydrodynamics and the coupling of a U(1) Goldstone mode to the fluid; initial conditions are treated as free inputs that must be chosen suitably. No new particles or forces are postulated.

free parameters (1)
  • initial hydrodynamic variables and Goldstone field values
    The attractor behavior and the value of the attractor time are stated to depend on the choice of suitable initial conditions.
axioms (1)
  • domain assumption Müller-Israel-Stewart theory provides a valid effective description of the hydrodynamic modes coupled to the Goldstone field
    The entire analysis is performed inside this framework; the abstract invokes it to model the out-of-equilibrium evolution.

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