Apparent bistability from weak long-range interactions

Achilleas Lazarides; Andrea Pizzi

REVIEW 5 major objections 4 minor 1 cited by

Apparent bistability from weak long-range interactions

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Critical droplet size diverges as $R_c \sim h^{-1/(\alpha-d)}$, so one stable phase looks like two.

desk verdict A likely-correct mechanism for apparent bistability with two independent analytic derivations, but the numerics never directly test the load-bearing scaling exponent and the conclusion misstates the lifetime. read the letter →

arxiv 2506.10068 v1 pith:2D7T7ZZM submitted 2025-06-11 cond-mat.stat-mech

Achilleas Lazarides , Andrea Pizzi This is my paper · ORCID

classification cond-mat.stat-mech MSC 82B2682C2082C27 PACS 05.70.Fh05.50.+q64.60.Q

keywords apparentbistabilitylong-rangeinteractionscriticaldropletnucleationmetastabilityprobabilisticcellularautomatonKacnormalizationphasecoexistence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that systems with power-law interactions $1/r^\alpha$ and a bias field $h$ can appear bistable, showing two practically stable phases, even when only one phase is truly stable. The mechanism is a critical droplet radius $R_c \sim h^{-1/(\alpha-d)}$ in the weak long-range regime $d<\alpha

What carries the argument

The carrying object is the critical droplet radius $R_c$, the radius at which the destabilizing bias $h$ balances the curvature-induced effective boundary field $n_R=-(f_d(\alpha)/R)^{\alpha-d}$ created by Kac-normalized power-law interactions. The argument uses two complementary routes: a coarse-grained Landau-Ginzburg energy balance in which the long-range wall energy density scales as $\ell^{-1/(\alpha-d)}$, and an explicit droplet integral giving the same exponent. Both routes lead to $R_c\sim h^{-1/(\alpha-d)}$ and to the apparent transition at $\alpha_c$.

What would settle it

Measure $R_c$ in $d=2$ for $\alpha=2.5$ with $h$ spanning two decades, say $0.1$ to $0.01$: if $R_c$ does not grow as $h^{-2}$, Equation (2) fails. Alternatively, run the cellular automaton at $L=10^4$ with $\alpha$ just below the apparent $\alpha_c$ and watch for a finite-time flip to the stable phase; if such a flip occurs on accessible timescales, the apparent bistability is merely finite-size, not practically stable.

Watch

Extended reading notes

Core claim

The central claim is that in $d<\alpha<d+1$, the effective boundary field on a droplet of radius $R$ is $n_R=-[f_d(\alpha)/R]^{\alpha-d}$, so balancing it against the bias $h$ gives $R_c=f_d(\alpha)\,h^{-1/(\alpha-d)}$. Because the exponent diverges as $\alpha\to d^+$, the critical radius can exceed the system size $L$ even for moderate $h$; any droplet that coarsening can generate is then subcritical, and the system remains in the metastable state for exponentially long times. The paper calls this apparent bistability: there is a unique stable phase in the thermodynamic limit, but for $\alpha<\alpha_c$ with $\alpha_c\approx d+\log(1/h)/\log(\beta L/f_d)$, the system is bistable for all practical purposes and $\alpha_c$ behaves like a genuine critical point, sharpening with $L$.

Load-bearing premise

The argument assumes nucleation is controlled by a single sharp, circular, uniform droplet whose boundary field is computed at one surface point and whose ultraviolet divergence is cut off at $|r|\geq 1$; if interface fluctuations, diffuse walls, or finite-size corrections change this barrier, the divergence of $R_c$ and the apparent transition would be altered.

Editorial extensions

If this is right

For any finite system size $L$ there is a range $d<\alpha<\alpha_c$ where the metastable state has an exponentially long lifetime, so apparent bistability is unavoidable in practice.
The apparent phase boundary $\alpha_c$ shifts only logarithmically with system size, so even very large simulations or experiments will see a sharp-looking transition.
Coarsening from a mixed initial state cannot nucleate a supercritical droplet for $\alpha<\alpha_c$, so the system falls back to the metastable phase.
The distinction between mathematical and physical stability is real: weak long-range interactions produce practically indistinguishable bistability without true phase coexistence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same $R_c\sim h^{-1/(\alpha-d)}$ scaling should appear in asynchronous Glauber dynamics of a long-range Ising model; testing it requires large sizes because the predicted $\alpha_c$ drift is logarithmic.
The mechanism could protect ordered phases in driven or prethermal systems, where bias-like fields are unavoidable, making apparent bistability a resource for time-crystalline order.
A quantitative prediction: the metastable lifetime should grow roughly as $\exp(h^{-d/(\alpha-d)})$, a signature that could be measured directly in a single-droplet experiment.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

David,

Short version: the central scaling R_c ~ h^{-1/(alpha-d)} for d < alpha < d+1 is derived two independent ways (field-theoretic energy balance and an explicit droplet integral) and matches the earlier d=1 results and the d=2 numerics from Horikx. The genuinely new content is the general-d derivation, the closed-form f_d(alpha) prefactors, and the framing of apparent bistability with a logarithmically drifting alpha_c. I think the mechanism is right and worth taking seriously, but the evidence is thinner than the abstract claims.

What works: the two analytic routes agree; the droplet calculation in Appendix B is concrete and yields finite f_d for alpha -> d+; and the CA simulations show a crossover at alpha_c ~ 2.55 that sharpens as L grows from 50 to 400. For a single h = 0.3 and T = 0.1 that is not a lot of data, but the trend is consistent with the mechanism. The update rule preserves detailed balance, which is the right choice when studying metastability.

The soft spots, in order of seriousness. First, the conclusion states that the metastable lifetime scales algebraically as h^{-1/(alpha-d)}. That is wrong on the paper's own terms: nucleation is exponential in R_c^d, so tau ~ exp(c h^{-d/(alpha-d)}). The body is consistent with exponential, so this is arguably a careless sentence, but it will mislead readers. Second, the h-dependence of R_c is never directly tested. All the numerics use one bias value; the exponent -1/(alpha-d) rests on the analytic arguments and prior work. Since the entire apparent-bistability picture hangs on that exponent, a direct R_c vs h measurement at fixed alpha is the natural missing figure. Third, the claim that alpha_c behaves like a genuine critical point is supported by the sigma_M peak, not by a finite-size scaling analysis (Binder cumulant or similar). With L up to 400, the sharpening is suggestive but not conclusive. Fourth, no code or data are provided, and the quantitative boundary depends on a hand-picked beta. The sharp-interface droplet assumption could modify prefactors but I doubt it changes the exponent, because the energy-balance derivation is more general.

Overall: the core concept is solid and likely correct, but the paper overreaches in the conclusion and underdelivers on direct verification. I would send it to referees: it deserves a serious look, and the referees should push for the h-sweep and a corrected conclusion. It is also a good reading-group paper: the mechanism is more interesting than the current numerical support.

Referee Report

5 major / 4 minor

Summary. The paper studies metastability in Ising-type systems with power-law interactions 1/r^alpha. For the weak long-range regime d < alpha < d+1, it derives via two complementary field-theoretic arguments that the critical droplet radius scales as R_c ~ h^{-1/(alpha-d)}, an exponent that diverges as alpha -> d+. The authors then argue that in a finite system of size L, coarsening from a mixed initial state produces stable droplets of size roughly beta L, and if R_c exceeds that size, the system appears bistable for practical purposes even though a unique stable phase exists in the thermodynamic limit. This yields an apparent critical exponent alpha_c ~ d + log(1/h)/log(beta L/f_d), which drifts logarithmically slowly with L. The claims are supported by numerical simulations of a probabilistic cellular automaton in d=2, showing a crossover in the long-time magnetization and a peak in its standard deviation that sharpen with L at alpha_c ~ 2.55 for h = 0.3.

Significance. If the central scaling and the apparent-bistability mechanism hold, the paper introduces a conceptually interesting distinction between mathematical and physical stability in long-range systems, with potential consequences for metastability and nucleation theory. A clear strength is that the droplet-size scaling is derived from two independent approaches (a coarse-grained field-theoretic scaling argument and an explicit sharp-interface droplet calculation), and the numerical automaton is designed to preserve detailed balance, avoiding known irreversibility artifacts. The paper also makes falsifiable predictions, such as the specific h-dependence of R_c and the logarithmic drift of alpha_c with L. However, as detailed below, several of these predictions are not directly tested, and two stated results (the coarsening energy scaling and the metastable lifetime scaling) appear incorrect as written. The overall idea is promising and the central mechanism is plausible, but the current evidence is insufficient to establish the claim convincingly.

major comments (5)

[Field theoretical insights (main text) and Appendix A] The stated coarsening energy-density scaling is internally inconsistent. The text says that for d < alpha < d+1, rho_E ~ ell^{-1/(alpha-d)}, but balancing rho_E ~ rho_h ~ h then gives ell_c ~ h^{-(alpha-d)}, not h^{-1/(alpha-d)} as claimed. The correct Bray scaling, consistent with Appendix A where the excess droplet energy is R^{d-sigma} with sigma = alpha-d, is rho_E ~ ell^{-(alpha-d)}. As written, the derivation of the central scaling law does not follow from the stated power counting; this needs to be corrected.
[Conclusion] The statement that 'the lifetime of the metastable phase scales as h^{-1} for alpha > d+1, and as h^{-1/(alpha-d)} for d < alpha < d+1' is incorrect. The paper's own nucleation argument gives t ~ exp(O(R^d)), so with R_c ~ h^{-1/(alpha-d)} the lifetime is exponential in h^{-d/(alpha-d)}, not a power law. The confusion appears to conflate the critical radius scaling with the lifetime; this should be corrected or the lifetime definition clarified.
[Numerics, Fig. 3] The central prediction R_c ~ h^{-1/(alpha-d)} is never directly tested. Figure 3(a) shows R_c versus alpha for a single bias h = 0.3, and no h-dependence of R_c is reported anywhere. Since this exponent is the basis of the apparent-bistability mechanism and of the expression for alpha_c, a direct measurement of R_c as a function of h for fixed alpha (and a comparison with the predicted exponent) is essential to support the claim.
[Numerics, Fig. 3(d)-(f)] The claim that alpha_c behaves like a genuine critical point is supported only by inspection of the crossover in the average magnetization and the peak in sigma_M for L = 50, 100, 200, 400. No finite-size scaling analysis (e.g., scaling collapse of the order parameter or the Binder cumulant) is performed, and no error bars are given. Moreover, the predicted logarithmic drift of alpha_c with L is not demonstrated; the data appear to show a sharpening without a measurable shift, but this is not quantified. Given that the mechanism relies on the slow drift, this should be explicitly verified.
[Field theoretical insights (Eq. for alpha_c)] The apparent boundary alpha_c ~ d + log(1/h)/log(beta L/f_d) depends on an undetermined parameter beta, the droplet fraction from coarsening, which is neither measured nor estimated in the paper. For the numerical value alpha_c ~ 2.55 (h = 0.3, L = 50 - 400) and f_d ~ 1, the formula with beta ~ O(1) gives alpha_c ~ 2.2 - 2.3, which is outside the quoted crossover. The paper should either measure beta from the numerics, or discuss the discrepancy and the sensitivity of alpha_c to beta.

minor comments (4)

[Abstract and Fig. 1(c)] The phrase 'exponential scaling of the critical droplet size R_c ~ h^{-1/(alpha-d)}' is imprecise: this is a power law in h, with an exponent that diverges as alpha -> d+. Please rephrase to avoid confusion.
[Appendix A, Eq. (A4)] The notation '+epsilon h R^d ... - h phi_0 R^d' is confusing because both terms are proportional to h R^d; the net bulk contribution should be defined once, with its sign, so that the balance with the surface terms is clear.
[Fig. 3(a) and footnote [44]] The caption states that R_c is obtained by preparing a droplet and checking when its magnetization stays constant, but the exact criterion is not given. The footnote about the periodic array of droplets is also unclear; please clarify the simulation protocol.
[Appendix B] The treatment of the ultraviolet cutoff (restricting |r| >= 1) and the claim that the neglected corrections in the d=2 integral are subleading for R >> 1 would benefit from a more explicit justification, as the scaling result depends on the cutoff being irrelevant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central R_c scaling is derived from the stated Hamiltonian and droplet geometry, and the numerical alpha_c is an independent dynamical observable, not a fitted restatement of the theory.

full rationale

The central exponent is derived, not fitted. Appendix A obtains Delta E ~ sigma R^{d-1} + gamma R^{d-sigma} (Eq. A2) and balances it against h phi_0 R^d (Eq. A4) to get R_c ~ h^{-1/sigma} with sigma = alpha - d; Appendix B computes the boundary effective field n_R by exact geometric integrals (Eqs. B2-B8) and sets n_{R_c} = -h, giving Eq. (2). Both derivations use the stated power-law Hamiltonian or Kac-normalized lattice model, so the prediction is an output of the model, not a restatement of the simulation outcomes. The numerical alpha_c ~ 2.55 is defined by long-time magnetization averages and standard-deviation peaks (Fig. 3d-f), not by imposing the theoretical condition R_c ~ beta L; therefore the apparent transition is an independent numerical finding. The constant beta in alpha_c ~ d + log(1/h)/log(beta L/f_d) is an explicitly phenomenological droplet-size factor, not a parameter fitted to the quantity being predicted; it affects the quantitative location of the boundary but does not make the derivation circular. Self-citations (Refs. 31-35, 47, 49, 50) appear only in contextual time-crystal remarks and are not load-bearing. Two non-circular weaknesses should be noted: the paper does not directly measure R_c(h) to test the exponent 1/(alpha-d), since Fig. 3(a) uses a single h = 0.3, and the conclusion's claim that the metastable lifetime scales as h^{-1/(alpha-d)} conflicts with the paper's own nucleation estimate t ~ e^{O(R^d)}; both are validation or correctness gaps, not circular reductions.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central scaling is derived, not fitted, but the quantitative boundary α_c depends on the hand-chosen constant β and on modeling assumptions (sharp interface, Kac normalization, UV cutoff, specific cellular automaton). No new physical entities are introduced.

free parameters (1)

β (droplet fraction from coarsening) = order 1, not determined
In the estimate α_c ≈ d + log(1/h)/log(βL/f_d), the typical size of droplets produced by coarsening is written as R ≈ βL with β an unspecified constant. It is not fitted, but it is chosen by hand and controls the quantitative estimate of α_c.

assumptions (4)

domain assumption The probabilistic cellular automaton with flip probability p(n_i)=1/2(1+tanh((n_i+h)/T)) satisfies detailed balance and is in the Ising universality class.
Stated in the model section (Eq. 4 and surrounding text); the transferability of the results to Hamiltonian settings (e.g., Glauber dynamics of a long-range Ising model) is explicitly left to future work.
domain assumption The coarse-grained Hamiltonian H = ∫[(1/2)(∇φ)^2 + V(φ)] + V_NL[φ] with V_NL[φ] = ∫∫ [φ(x)-φ(x')]^2/|x-x'|^{d+σ} describes the universality class of the spin models.
Invoked in Appendix A, following Bray [10]; the droplet excess energy ∆E ~ R^{d-1} + γR^{d-σ} relies on this Hamiltonian.
domain assumption The Kac normalization N_α with the UV cutoff |r|≥1 removes the divergence and defines the model; the droplet calculation then treats space as continuous.
Used in Eq. (3) and Appendix B; the cutoff at |r|=1 is an arbitrary choice, and the continuous-space treatment is justified only for large critical droplets.
ad hoc to paper Coarsening from a mixed initial state generates stable droplets of a typical size R ≈ βL with β an O(1) constant, and nucleation over the sharp-interface barrier is the only escape route from the metastable phase.
This assumption appears in the heuristic derivation of α_c (paragraph after Eq. (2)) and is the basis for the claim that the system appears bistable for α < α_c.

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Pith. "Pith review of Apparent bistability from weak long-range interactions." pith.science (2026). https://pith.science/paper/2D7T7ZZM

@misc{pith2026250610068,
  author       = {Pith},
  title        = {Pith review of: Apparent bistability from weak long-range interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2D7T7ZZM}},
  note         = {Machine review of arXiv:2506.10068}
}

abstract

Bistability, or the coexistence of two stable phases, can be broken by a bias field $h$ destabilising one of the phases via the nucleation and growth of defects. Strong long-range interactions, $1/r^\alpha$ with $\alpha$ less than the system's dimensionality $d$, can suppress the proliferation of defects and restore bistability. The case of weak long-range interactions $d<\alpha < d+1$ remains instead poorly understood. Here, we show that it supports \emph{apparent} bistability: While the system has in principle a unique stable phase, it appears bistable for all practical purposes for $\alpha < \alpha_c$, with $\alpha_c > d$ behaving like a genuine critical point. At the core of this is an exponential scaling of the critical droplet size $R_c\sim h^{-1/(\alpha - d)}$, which makes nucleating destabilizing droplets extremely unlikely for $\alpha < \alpha_c$, and such that $\alpha_c$ is mostly independent of system size. In support of these conclusions we provide field-theoretical arguments and numerics on a probabilistic cellular automaton. Overall, our results offer a way to rethink phase stability in systems with long-range interactions as well as a new route to achieve practical bistability.

Figures

Figures reproduced from arXiv: 2506.10068 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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**Figure 4.** Figure 4: FIG. 4. Graphical representation of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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