The Lee--Yang Edge Exponent via Logarithmic Averaging

Abstract

Let $F$ be the thermodynamic free energy of a ferromagnetic Ising model,analytic on $\mathbb{C}^{*}\setminus\mathcal{Z}_{\beta}$. 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.