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Functional Dimensional Regularization for the Lee-Yang Model

Piero Beretta, Carlos A. Sánchez-Villalobos, Alessandro Codello

Nov 2, 2026arXiv:2611.03318v1
hep-thcond-mat.stat-mech
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Scorecard· 16/16
4.5/10 impact

Competent, honest method-validation of FDR on Lee-Yang with good d=4,5 agreement, but incremental, not precision-competitive, and weak in d=3.

Abstract

We apply functional dimensional regularization (FDR) to the Lee-Yang universality class, a paradigmatic non-unitary theory, and compute its main critical exponents across the dimension range 3≤d≤63\leq d\leq 6. By combining a second-order derivative expansion with polynomial field expansions, we obtain results that are compatible with resummed ε\varepsilon-expansion as well as with state-of-the-art estimates from Conformal Bootstrap and Fuzzy Sphere techniques. This demonstrates that, in the context of non-unitary theories, FDR retains the efficiency and rapid convergence to accurate critical exponents already observed in the Ising and O(N) cases.

AI Impact Assessments

(1 model)

Core Contribution

The paper applies Functional Dimensional Regularization (FDR), a recently proposed functional RG scheme from the same authors, to the Lee-Yang (Yang-Lee edge singularity) universality class. This is a non-unitary, PT-symmetric φ³ theory with upper critical dimension 6. Using a second-order derivative expansion (DE2) with polynomial field expansions, they compute η, ν, σ, Δφ, and parts of the RG spectrum for 3 ≤ d ≤ 6. Two expansions are used: around zero field and around the zero-mass point (v''=0). The flow equations are closed-form and dimension-agnostic, with d entering only through classical scaling.

They recover the one-loop ε-expansion analytically (η = −ε/9, ν = 1/2 + 5ε/36, ω = ε). They also give the first functional-RG estimates of the correction-to-scaling exponent ω (and Δφ³) in d = 5, 4, 3. They analyze the scaling-relation violation Δν and attribute the failure of the zero-mass expansion to spurious mixing with redundant operators.

The novelty is mostly in the application: the FDR machinery was developed in earlier papers on Ising and O(N), so this is a "testing campaign" entry. The scientific content is mainly a stress test of FDR on a non-unitary theory with complex couplings.

Methodological Rigor

  • The approach is sound and follows established NPRG practice (An et al., Zambelli–Zanusso). The PT-symmetric parametrization keeps all couplings real.
  • The ε-expansion recovery is a good analytic check. The cross-check with the φ³ beta functions at d = 6 is also useful.
  • Convergence with truncation order (up to degree 20) is shown, and the comparison table is broad: six-loop resummed εE, CB, FS, HT, and NPRG-DE2.
  • The scaling-violation diagnostic Δν is honest and informative. The authors report that the zero-mass expansion, which converges fastest, still violates scaling by about 8% in d=5, 15% in d=4, and over 50% in d=3. They say the d=3 approximation is "inadequate" for the spectrum.
  • Weaknesses:

    • - No systematic error estimates are given; the table omits errors.

    - The abstract's "compatible with" claim is loose. In d=3, η = −0.545 (FDR) against −0.57 to −0.58 from the best methods, and ν differs noticeably.

    - The agreement in d=4, 5 is good, but the large scaling violation raises doubts about how reliable the d=3 numbers are.

    - The claim of "rapid convergence" is partly undermined by the Δν saturation.

    - The ω estimates are contaminated by redundant-operator mixing, and the authors acknowledge this.

    Potential Impact

  • It extends FDR's reach to a non-unitary theory and shows that the framework is at least viable there. This supports FDR as a lighter-weight alternative to NPRG, with simpler flow equations that allow higher-order field expansions.
  • The explanation of the scaling-violation problem through redundant operators, and the pointer to Essential RG, is a useful diagnostic. It may motivate follow-up work.
  • The first functional-RG ω estimates are a modest but concrete data point.
  • The broad field impact is limited. Lee-Yang exponents are already known with decent precision from six-loop ε-expansion, FS, and CB, and this paper does not beat them. It is not a precision result. The value is in validating a method, not in new physics.
  • Planned extensions to the tricritical Lee-Yang and other non-unitary multicritical models, and to the edge-singularity location, could be more impactful. This paper is a stepping stone.

    • Timeliness & Relevance

    The timing is good. Fuzzy-sphere results for Yang-Lee (2025), the six-loop ε-expansion, and renewed interest in non-unitary CFTs and QCD critical-point searches make cross-method benchmarking in d = 3–6 relevant. The high-order ε-expansion and FS comparisons make the paper timely. The paper is, however, a specialized contribution in the FRG community.

    Strengths

  • It gives a continuous-in-d study of the full spectrum.
  • It checks against the analytic one-loop result.
  • It compares systematically against multiple methods.
  • It candidly reports failures: the LPA' inconsistency, redundant-operator mixing, and the large d=3 scaling violation.
  • The comparison table is useful for readers, and the note on the CB two-solution ambiguity and the FS d=4 convergence is helpful.

    • Limitations

  • It is incremental relative to the FDR Ising/O(N) papers.
  • It gives no error bars and no quantitative convergence criterion in the comparison table.
  • The d = 3 results are the least trustworthy, and the zero-field expansion fails to converge there.
  • The spectrum has spurious level repulsion, and the higher operators are unreliable.
  • The method is not yet competitive in precision with ε-expansion resummation or FS.
  • The text has some typographical oddities (the reference list duplicates entries, and the arXiv IDs and dates look odd).
  • No code is released, although the equations are explicit enough to implement.

    • Other Observations

  • The paper is reproducible in principle: the flow equations are given in closed form, the expansions are specified, and the fixed-point search procedure is described. The ansatz is simple enough for a competent FRG practitioner to implement in Mathematica in weeks.
  • Computational cost is low, in line with a small theory group.
  • The result that a running η must be included in the stability matrix to obtain the correct O(ε) result is a neat technical observation specific to theories where η ~ ε.

    • Overall this is a competent, honest, moderately useful method-validation paper with limited standalone influence and mainly incremental impact on FRG methodology.

    Rating:4.5/ 10
    Significance 4Rigor 5.5Novelty 3.5Clarity 6

    Generated Nov 3, 2026

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