The Lee--Yang Edge Exponent via Logarithmic Averaging
Qiao Wang
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 $h_φ<0$ 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.