The Lee--Yang Edge Exponent via Logarithmic Averaging

TL;DR

This paper investigates the Lee--Yang edge exponent in ferromagnetic Ising models, establishing connections between logarithmic averaging, monodromy, and conformal field theory properties.

Contribution

It introduces new theorems linking the Lee--Yang edge exponent to Jensen averages, monodromy, and RG fixed points, providing unconditional results for the 2D Ising model.

Findings

Jensen average derivative equals the edge exponent plus one.

Monodromy around the edge multiplies the singular part by a root of unity.

Edge expansion follows from density asymptotics via Mellin transform.

Abstract

Let $F$ be the thermodynamic free energy of a ferromagnetic Ising model,analytic on $\mathbb{C}^{*}\setminus\mathcal{Z}_{\beta}$. The Lee--Yang edge at $z_c\in\partial\mathcal{Z}_\beta$ is characterised by $F(z)=F(z_c)+B(z-z_c)^{\sigma+1}+o(|z-z_c|^{\sigma+1})$ with $\sigma\in(-1,0)$ and $B\neq 0$. We prove three results: Theorem A (Jensen slope): defining the Jensen average $\widetilde{N}(x)=\frac{1}{2\pi}\int_0^{2\pi}\log|\widetilde{F}(e^{x+i\theta})|\,d\theta$ of $\widetilde{F}=F-F(z_c)$, the edge exponent satisfies $\widetilde{N}'(0^+)=\sigma+1$. 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 $e^{2\pi i(\sigma+1)}$, a primitive $q$-th root of unity when $\sigma+1=p/q$. Theorem C (Kac monodromy): for any 2D CFT at an RG fixed point with relevant operator $\phi$ of weight $h_\phi<0$ satisfying the Lee--Yang property, the RG scaling equation forces $\sigma=h_\phi/(1-h_\phi)$ and monodromy order $q=\mathrm{denom}(1/(1-h_\phi))$. We also prove that the edge expansion follows from the density asymptotics $\rho(\theta)\sim A|\theta-\theta_c|^\sigma$ via a Mellin-transform calculation, making all three theorems unconditional for the $d=2$ Ising model.