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REVIEW 2 major objections 6 minor 58 references

Long-range Ising interactions split false-vacuum decay into two regimes, with lifetime exponent B ~ h^{-1/\sigma} when \sigma < 1.

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2026-07-12 03:36 UTC pith:5B4MPIAJ

load-bearing objection Clean first field-theoretic treatment of Coleman nucleation with a fractional spatial kernel; two-regime B(h) scalings hold up under full nonlocal numerics. the 2 major comments →

arxiv 2607.03274 v1 pith:5B4MPIAJ submitted 2026-07-03 quant-ph gr-qchep-th

False vacuum decay in long-range interacting quantum systems

Valerio Pagni , Laura Batini , Nicol\`o Defenu This is my paper
classification quant-ph gr-qchep-th
keywords false vacuum decaylong-range interactionsbouncefractional Laplacianthin-wall approximationquantum simulatorsIsing chainbubble nucleation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper maps a mixed-field Ising chain with power-law couplings onto a nonlocal Euclidean field theory whose spatial kinetic term is a fractional Laplacian. The resulting critical bubble (the bounce) is anisotropic and carries algebraic spatial tails, so the usual compact thin-wall picture no longer looks automatic. Thin-wall analysis plus full numerical solution of the nonlocal saddle nevertheless show that the leading exponents that control the lifetime survive: for 0 < \sigma < 1 the bounce action scales as B ~ h^{-1/\sigma}, while for 1 < \sigma < 2 one recovers the ordinary Coleman scaling B ~ h^{-1} with a long-range correction ~ h^{\sigma-2}. Because the same platforms that engineer long-range Ising models already observe bubble nucleation, the interaction range becomes a direct experimental knob for the lifetime of metastable entangled states.

Core claim

Algebraic spatial tails of the nonlocal bounce leave the leading thin-wall exponents intact. Consequently the lifetime exponent that governs false-vacuum decay obeys B ≃ C1 h^{-1/\sigma} + C2 h^{-1} for 0 < \sigma < 1 and B ≃ C1 h^{-1} + C2 h^{\sigma-2} for 1 < \sigma < 2, with the crossover at \sigma = 1 separating infrared-dominated from ultraviolet-dominated nucleation.

What carries the argument

Anisotropic Derrick identities (T\tau/Kx = \sigma/2, U/Kx = (\sigma-2)/2) together with the thin-wall crossover equation \epsilon = \beta R_x^{-\sigma} + \gamma R_x^{-1} that fixes the critical radius and therefore the action scaling B \propto R_x.

Load-bearing premise

That the continuum action with a pure fractional Laplacian and a simple tilted double-well potential fully captures the universal nucleation physics of the original lattice model for every range 0 < \sigma < 2.

What would settle it

In a trapped-ion mixed-field Ising chain with tunable power-law exponent \alpha (hence \sigma = \alpha-1), measure the false-vacuum lifetime versus longitudinal bias h; the extracted exponent of B(h) must equal 1/\sigma below \sigma = 1 and 1 above it.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For \sigma < 1 the critical bubble size and lifetime grow steeply as the bias vanishes, strongly suppressing nucleation relative to local systems.
  • For 1 < \sigma < 2 the leading lifetime is still Coleman-like, yet the bubble remains anisotropic and algebraically tailed, giving a real-space signature of nonlocality.
  • The boundary where long-range effects dominate nucleation is shifted from the equilibrium value \sigma* = 2-\eta down to \sigma = 1.
  • Interaction range becomes a laboratory control parameter for the lifetime of metastable highly entangled states in quantum simulators.

Where Pith is reading between the lines

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

  • Near \sigma = 0 the dilute-bounce picture may break down because simultaneous nucleation of multiple subcritical bubbles can trigger avalanche growth through long-range interactions.
  • Matching the continuum stiffnesses and potential to the microscopic spin-chain spectrum would convert the present scaling theory into parameter-free lifetime predictions for existing trapped-ion experiments.
  • The same fractional-kernel construction should apply to two-dimensional Rydberg arrays, predicting an analogous IR/UV crossover once the appropriate σ is identified.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper maps false-vacuum decay in a long-range mixed-field Ising chain (1/r^α couplings) to a spatially nonlocal Euclidean φ^4 theory with fractional spatial kinetic term |q|^σ (σ=α−1). It shows that the critical bounce is anisotropic and develops algebraic spatial tails φ∼|x|−(1+σ). Combining a thin-wall analysis with numerical solutions of the full nonlocal saddle, the authors derive and confirm a two-regime scaling of the bounce action B that controls the lifetime: B∼h−1/σ for 0<σ<1 (IR-dominated) and B∼h−1 plus a subleading ∼h^{σ−2} correction for 1<σ<2 (UV-dominated). Exact anisotropic Derrick identities are derived and verified numerically; free-exponent fits to the full saddles recover the predicted leading exponents.

Significance. If correct, the result supplies a concrete, tunable control parameter (the interaction range σ) for false-vacuum lifetimes in quantum simulators that already engineer power-law Ising couplings (trapped ions, Rydberg arrays, cavity systems). The two-regime structure, the survival of thin-wall exponents despite algebraic tails, and the anisotropic Derrick identities are non-obvious and experimentally relevant. Strengths include transparent thin-wall bookkeeping that recovers exact scaling identities, a full nonlocal PDE solver (Gaussian basis + Chebyshev collocation + Newton–Krylov continuation) with residuals <10−9 and Derrick ratios verified to ≲2%, and free-exponent numerical confirmation of the predicted ϑ(σ). These elements make the central claim falsifiable and useful beyond pure theory.

major comments (2)
  1. End Matter (continuum mapping) and the statement that Eq. (2) is treated as the effective continuum theory: the pure |q|^σ + phenomenological tilted φ^4 truncation is standard, but the manuscript should state more sharply when residual analytic q^2 stiffness and non-polynomial lattice potential features remain irrelevant at the critical radii R_x(h) actually accessed. For σ→2− or moderate h the healing length and R_x can approach lattice scales; a short estimate of the window of h and σ where the leading exponents are protected would make the continuum claim load-bearing rather than assumed.
  2. Main-text Eq. (8) and Fig. 3: the free-exponent fits recover ϑ=1/σ (σ<1) and ϑ=1 (σ>1), which is strong support. However, the accessible h range is still pre-asymptotic (subleading terms are essential, as the SM pure-power-law comparison shows). The paper should report the fitted C1, C2 (with uncertainties) and the h window used for each σ so that the claimed leading exponents can be independently re-fit and the size of pre-asymptotic contamination quantified.
minor comments (6)
  1. Fig. 1 caption and main text: the algebraic tail is written ∼|x|−(1+σ); a brief derivation or pointer to the linearised nonlocal equation (or to the companion note [40]) would help readers who do not immediately see why the exponent is 1+σ rather than σ.
  2. Eqs. (4)–(5) and Fig. 2: the Derrick ratios are an excellent diagnostic; stating the maximum relative deviation over the full (σ,h) set in the main text (not only in the SM) would strengthen the numerical claim for readers who skip the supplement.
  3. Thin-wall section, Eq. (7): the signs of β and γ for σ ≶ 1 are explained in the End Matter; a one-sentence reminder in the main text would avoid confusion when the IR term dominates for σ<1.
  4. Conclusions: the discussion of possible multi-bubble/avalanche physics for σ→0 is interesting but speculative; a clearer separation between the controlled single-bounce results and open questions would improve focus.
  5. Notation: c_τ, c_LR are set to 1 after Eq. (2); stating once that all dimensionful quantities are measured in these units (and that only ratios of scales enter the exponents) would remove any ambiguity for experimental matching.
  6. References: recent experimental bubble-nucleation works in Rydberg arrays and quantum annealers are cited; adding a short pointer to classical long-range droplet literature (beyond McCraw) would situate the IR-dominated regime more clearly.

Circularity Check

0 steps flagged

No significant circularity: thin-wall exponents and free-exponent numerical fits follow from an independently written nonlocal action and its saddle, not from self-definition or target-fitting.

full rationale

The derivation chain is self-contained. The continuum action (2) is obtained from a standard Suzuki–Trotter + Hubbard–Stratonovich mapping of the long-range Ising chain (End Matter); the fractional kernel |q|^σ is the known small-momentum expansion of the power-law interaction, not a quantity defined from the bounce action. Anisotropic Derrick identities (4)–(5) are obtained by two independent dilations of that action and are verified a posteriori on the numerical saddles. The thin-wall crossover equation (7) and the two-regime scalings (8) follow by evaluating the same action on a compact-droplet ansatz and balancing the IR fractional wall energy against the UV core; the leading powers are therefore predictions of the action, not inputs. Numerical confirmation solves the full nonlocal PDE (3) by Newton–Krylov continuation and then fits B(h) with free leading exponent ϑ (Eq. (9) and Fig. 3 inset), recovering ϑ = 1/σ (σ < 1) and ϑ = 1 (σ > 1). Self-citations supply background on long-range criticality and the fractional Laplacian but do not enter the bounce-action derivation or force the exponents. The continuum truncation (pure |q|^σ + phenomenological ϕ^4) is an assumption about universality, not a circular step. Score 1 reflects only the ordinary presence of author-overlap background citations that are not load-bearing for the central claim.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central claim rests on a standard continuum mapping of the long-range Ising chain plus a thin-wall plus numerical analysis of the resulting nonlocal saddle. Free parameters are only the overall units chosen for convenience; the axioms are the usual long-wavelength and thin-wall assumptions of the field. No new particles or forces are postulated.

free parameters (2)
  • c_τ = c_LR = g = ϕ_0 = 1
    Overall stiffnesses and potential scales set to unity; non-universal and can be restored by matching to a concrete lattice spectrum, but the reported exponents are independent of these choices.
  • Gaussian basis width q = w_g Δx = 0.60
    Numerical hyper-parameter of the spatial discretization; convergence is checked via Derrick identities, yet the precise value is chosen by hand.
axioms (4)
  • domain assumption Long-wavelength continuum limit of the power-law Ising chain yields a pure fractional Laplacian |q|^σ (σ=α-1) that dominates the analytic q^{2} term for 0<σ<2.
    Stated in the mapping section and End Matter; standard for long-range criticality but assumes that lattice-scale analytic terms remain irrelevant at the critical bubble radii.
  • domain assumption The effective potential may be replaced by a tilted ϕ^{4} double well without changing the universal nucleation exponents.
    Explicitly asserted after Eq. (2); justified by the requirement of two minima and a small energy splitting, but the microscopic Hubbard-Stratonovich potential is non-polynomial.
  • ad hoc to paper Thin-wall droplet ansatz (compact true-vacuum core of radii R_x, R_τ with fixed healing lengths) captures the leading and first sub-leading action scalings even in the presence of algebraic tails.
    Used to derive Eq. (8); subsequently validated numerically, yet remains an uncontrolled approximation a priori for heavy-tailed profiles.
  • standard math Anisotropic Derrick identities obtained from independent dilations in x and τ hold for the exact nonlocal bounce.
    Derived from stationarity under two independent scalings; verified numerically to high accuracy.

pith-pipeline@v1.1.0-grok45 · 22795 in / 2931 out tokens · 27799 ms · 2026-07-12T03:36:27.929724+00:00 · methodology

0 comments
read the original abstract

We formulate false-vacuum decay in a mixed-field Ising chain with $1/r^\alpha$ interactions as a spatially nonlocal Euclidean $\phi^4$ theory featuring a fractional spatial kinetic term $\sim |q|^\sigma$, where $\sigma=\alpha-1$. The nonlocal bounce is anisotropic in space-time and develops algebraic spatial tails, challenging the standard thin-wall picture of a compact droplet. Combining thin-wall arguments with numerical solutions of the full nonlocal saddle, we show that these tails preserve the leading thin-wall exponents, manifesting instead in subleading corrections. For $0<\sigma<1$, the lifetime exponent scales with the energy bias $h$ of the metastable state as $B\sim h^{-1/\sigma}$; for $1<\sigma<2$, the leading Coleman scaling $B\sim h^{-1}$ is recovered, while long-range effects are retained in the subdominant term $\sim h^{\sigma-2}$. Our results show that tunable long-range interactions fundamentally reshape bubble nucleation and alter false-vacuum decay in quantum simulators.

Figures

Figures reproduced from arXiv: 2607.03274 by Laura Batini, Nicol\`o Defenu, Valerio Pagni.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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