The Lee--Yang Edge Exponent via Logarithmic Averaging

Abstract

Let $F$ be the thermodynamic free energy of a ferromagnetic Ising model,analytic on ${\mathbb{C}}^{\ast }\setminus {\mathcal{Z}}_{\hat{I}}^2$. The Lee--Yang edge at $z_c\in \partial {\mathcal{Z}}_{\hat{I}}^2$ is characterised by $F(z)=F(z_c)+B(z-z_c{)}^{\ddot{I}ƒ+1}+o(|z-z_c{|}^{\ddot{I}ƒ+1})$ with $\ddot{I}ƒ\in (-1,0)$ and $B\ne 0$. We prove three results: Theorem A (Jensen slope): defining the Jensen average $\tilde{N}(x)=\frac{1}{2\ddot{I}€}{\int }_0^{2\ddot{I}€}\log |\tilde{F}(e^{x+i\hat{I}¸})|d\hat{I}¸$ of $\tilde{F}=F-F(z_c)$, the edge exponent satisfies ${\tilde{N}}^{\prime }(0^{+})=\ddot{I}ƒ+1$. 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 $e^{2\ddot{I}€i(\ddot{I}ƒ+1)}$, a primitive $q$-th root of unity when $\ddot{I}ƒ+1=p/q$. Theorem C (Kac monodromy): for any 2D CFT at an RG fixed point with relevant operator $\ddot{I}•$ of weight $h_{\ddot{I}}•<0$ satisfying the Lee--Yang property, the RG scaling equation forces $\ddot{I}ƒ=h_{\ddot{I}}•/(1-h_{\ddot{I}}•)$ and monodromy order $q=\mathrm{denom}(1/(1-h_{\ddot{I}}•))$. We also prove that the edge expansion follows from the density asymptotics $\ddot{I}(\hat{I}¸)\sim A|\hat{I}¸-\hat{I}{¸}_c{|}^{\ddot{I}}ƒ$ via a Mellin-transform calculation, making all three theorems unconditional for the $d=2$ Ising model.