The Lee--Yang Edge Exponent via Logarithmic Averaging

Abstract: Let $F$ be the thermodynamic free energy of a ferromagnetic Ising model,analytic on $\mathbb{C}^{*}\setminus\mathcal{Z}_β$. The Lee--Yang edge at $z_c\in\partial\mathcal{Z}β$ is characterised by $F(z)=F(z_c)+B(z-z_c)^{σ+1}+o(|z-z_c|^{σ+1})$ with $σ\in(-1,0)$ and $B\neq 0$. We prove three results: Theorem A (Jensen slope): defining the Jensen average $\widetilde{N}(x)=\frac{1}{2π}\int_0^{2π}\log|\widetilde{F}(e^{x+iθ})|\,dθ$ of $\widetilde{F}=F-F(z_c)$, the edge exponent satisfies $\widetilde{N}'(0^+)=σ+1$. The proof is a direct application of Jensen's formula. Theorem B (Monodromy): the monodromy of $F$ around $z_c$ multiplies the singular part by $e^{2πi(σ+1)}$, a primitive $q$-th root of unity when $σ+1=p/q$. Theorem C (Kac monodromy): for any 2D CFT at an RG fixed point with relevant operator $φ$ of weight $hφ<0$ satisfying the Lee--Yang property, the RG scaling equation forces $σ=h_φ/(1-h_φ)$ and monodromy order $q=\mathrm{denom}(1/(1-h_φ))$. We also prove that the edge expansion follows from the density asymptotics $ρ(θ)\sim A|θ-θ_c|^σ$ via a Mellin-transform calculation, making all three theorems unconditional for the $d=2$ Ising model.

• The paper introduces the Jensen slope formulation, linking the right derivative of a logarithmically averaged free energy near the Lee–Yang edge to the edge exponent.
• It derives monodromy properties showing that analytic continuation around the singularity produces a multiplier determined by the rational value of the exponent, exemplified in the 2D Ising model.
• It establishes a rigorous connection between renormalization group scaling in conformal field theory and the Lee–Yang edge, enabling unconditional calculations of critical exponents through density asymptotics.

The Lee--Yang Edge Exponent via Logarithmic Averaging

Overview

This paper rigorously investigates the Lee--Yang edge singularity in the context of the ferromagnetic Ising model. The principal contribution is a novel formulation of the edge exponent $\sigma$ via the slope of a Jensen-type average of the thermodynamic free energy. The work provides new, unconditional derivations of critical exponents, establishes the monodromy structure at the Lee--Yang edge, and demonstrates universal predictions for a wide class of conformal field theories (CFTs) at renormalization group (RG) fixed points displaying the Lee--Yang property.

Theoretical Foundations

The classical Lee--Yang theorem asserts that all zeros of the ferromagnetic Ising model partition function in the fugacity variable $z$ lie on the unit circle, and, in the thermodynamic limit, accumulate on an arc whose endpoints are termed the Lee--Yang edges. At these edges, the free energy develops a singularity characterized by a universal exponent $\sigma$ through the asymptotic expansion $$F(z) = F(z_c) + B(z-z_c)^{\sigma+1} + o(|z-z_c|^{\sigma+1})$$ with $z_c$ on the unit circle. The exponent $\sigma$ is dimension-dependent: it takes the exact value $-1/6$ for $d=2$ (as known from Virasoro minimal model $\mathcal{M}(2,5)$ and the thermodynamic Bethe ansatz), $1/2$ for mean-field ($z$0), and approximately $z$1 for $z$2.

Main Results

Jensen Slope Characterization (Theorem A)

A central innovation is the definition of the Jensen average $z$3 centered at the Lee--Yang edge. Theorem A demonstrates that the right derivative at $z$4 is precisely the Lee--Yang edge exponent plus one: $z$5. This follows from a careful application of Jensen's formula, and leverages the analytic structure of $z$6 without reliance on algebraic-geometric constructs such as Newton polytopes. The result provides a direct analytic observable for extracting the edge exponent from thermodynamic data.

Monodromy (Theorem B)

The paper establishes that analytic continuation of the free energy $z$7 around the edge singularity $z$8 induces a monodromy transformation on the singular part, multiplying it by $z$9. When $\sigma$0 is rational, the operation corresponds to a root of unity whose order equals the denominator of $\sigma$1 in lowest terms, which for the 2D Ising model is $\sigma$2.

Conformal Field Theoretic Predictions (Theorem C)

For 2D CFTs at RG fixed points with a relevant operator $\sigma$3 of conformal weight $\sigma$4, the paper proves that $\sigma$5, with monodromy order given by $\sigma$6. This result is independent of traditional field-theoretic treatments and is derived strictly from the RG scaling property of the singular free energy, together with the Lee--Yang condition. Explicit values are derived for the entire infinite series of minimal models $\sigma$7, providing a direct bridge between the scaling dimensions and analytic structure at the edge.

Derivation from Density Asymptotics

The paper proves that the edge singularity expansion is a consequence of the local density of zeros, $\sigma$8, through a Mellin transform argument, providing an explicit formula for the coefficient $\sigma$9 in terms of $$F(z) = F(z_c) + B(z-z_c)^{\sigma+1} + o(|z-z_c|^{\sigma+1})$$0 and $$F(z) = F(z_c) + B(z-z_c)^{\sigma+1} + o(|z-z_c|^{\sigma+1})$$1. This closes the logical structure, enabling all main theorems to be established unconditionally (in particular, for the $$F(z) = F(z_c) + B(z-z_c)^{\sigma+1} + o(|z-z_c|^{\sigma+1})$$2 Ising model).

Methodological Advancements

The analytic techniques presented are notably robust. By framing the Jensen slope as a universal observable, the calculation of exponents is reduced to log-averaging procedures well-suited to both analytic and possibly numerical approaches. The clean separation of the regular and singular parts of the free energy expansion, and the exact computation of monodromy, are of significant technical interest, as is the bypassing of algebraic geometric apparatus in favor of harmonic and complex analytic arguments.

Implications and Future Directions

The analytic identification of the edge exponent via logarithmic averaging offers a new, practical approach to computing critical exponents in statistical mechanics systems, with direct extensions to systems beyond the Ising universality class. The monodromy structure revealed has implications for the study of analytic continuation and Riemann surface structures arising in partition function analyses. The RG/CFT prediction for $$F(z) = F(z_c) + B(z-z_c)^{\sigma+1} + o(|z-z_c|^{\sigma+1})$$3 provides an explicit correspondence between CFT data (conformal weight) and singularity structure, suggesting avenues for the identification of critical behavior in non-integrable and higher dimensional models.

Potential future directions include the extension of these analytic techniques to quantum lattice systems, models with more complex zero loci, or situations where the edge expansion has additional logarithmic corrections. The framework also suggests possible links to logarithmic CFTs and models with nontrivial Stokes phenomena in their free energy.

Conclusion

This paper delivers a rigorous, analytic framework for understanding the Lee--Yang edge singularity, the associated edge exponent, and monodromy properties, with unconditional results for the two-dimensional Ising model and predictive power in two-dimensional conformal field theory. The methodology primes additional analyses in statistical physics, field theory, and the mathematics of critical phenomena.