Global complex-temperature phase diagram of the Curie–Weiss model: the critical curve conjecture of Shamis and Zeitouni

The Curie–Weiss (mean-field Ising) model admits a rigorous partial description of its free energy $F_\beta=\lim_{N\to\infty}N^{-1}\log|Z_{\beta,N}|$ when the inverse temperature $\beta$ is complex: Shamis and Zeitouni (2017) proved that $F_\beta$ vanishes in a wedge $c\sqrt{\epsilon}\le R\le c'/\sqrt{\epsilon}$ (for $\beta=1+\epsilon+iR$) and for $\epsilon<0$, that $F_\beta>0$ on one explicit curve, and gave a complete local description of the Fisher zeros near the critical point $\beta=1$. They conjectured that the zero and positive free-energy regions are separated by a single critical curve $\{1+\epsilon_0(R)+iR\}$ with $\epsilon_0(0)=0$, $\epsilon_0(R)>0$ for $R\neq0$, and $\lim_{R\to\infty}R^2\epsilon_0(R)=b\in(0,\infty)$, the curve being determined by the saddle-point branch $u^*(\beta)$ through $\mathrm{Re}\, f_\beta(u^*(\beta))=0$. Whether this global two-phase description holds, and the exact shape of the critical curve (including the constant $b$), remains open. The complex-temperature behavior outside a neighborhood of $\beta=1$ remains largely unknown (Kabluchko, 2022/2025).

Background

The Curie–Weiss model is the mean-field (complete-graph) Ising model of ferromagnetism: $N$ spins $\sigma_i\in\{-1,+1\}$ interact all-to-all with Hamiltonian $H_N(\sigma)=-(2N)^{-1}\sum_{i,j}\sigma_i\sigma_j$, and the partition function at inverse temperature $\beta$ is $Z_{\beta,N}=2^{-N}\sum_\sigma \exp(-\beta H_N(\sigma))$. For real $\beta>0$ the model is exactly understood: by the classical work of Ellis and Newman, $\lim_{N\to\infty}N^{-1}\log Z_{\beta,N}=0$ for $0<\beta\le 1$, while for $\beta>1$ the limit equals $\log\cosh(\beta m_*)-\beta m_*^2/2>0$, where $m_*\in(0,1)$ is the spontaneous magnetization, i.e. the positive solution of $m_*=\tanh(\beta m_*)$; the model exhibits a second-order phase transition at the critical inverse temperature $\beta=1$.

Since the work of Yang and Lee (1952), phase transitions are also analyzed through the zeros of $Z_{\beta,N}$ as a polynomial in a complexified control parameter: zeros of the partition function in the complex inverse-temperature plane are called Fisher zeros, and their accumulation set on the real axis locates phase transitions. Shamis and Zeitouni (2017) studied the Curie–Weiss model at complex temperature $\beta\in\mathbb{C}$ (zero external field), asking for the phase diagram of the limiting exponential growth rate of the absolute partition function, $F_\beta=\lim_{N\to\infty}\frac{1}{N}\log|Z_{\beta,N}|,$ i.e. the division of the $\beta$-plane into a region where $F_\beta=0$ and a region where $F_\beta>0$ (the boundary is the asymptotic locus of Fisher zeros). Their analysis is based on the Laplace-type representation $Z_{\beta,N}=\sqrt{\beta N/(2\pi)}\int e^{-Nf_\beta(u)}du=\sqrt{N/(2\pi\beta)}\int e^{-Nh_\beta(u)}du$ with $f_\beta(u)=\beta u^2/2-\log\cosh(\beta u)$ and $h_\beta(u)=u^2/(2\beta)-\log\cosh u$; the relevant saddles solve $f'_\beta(u)=\beta u-\beta\tanh(\beta u)=0$, i.e. $u=\tanh(\beta u)$, which defines a multivalued function $u(\beta)$, equivalently $\beta=(2u)^{-1}(\log\frac{1+u}{1-u}+2\pi i k)$, $k\in\mathbb{Z}$.

What is proved (Shamis–Zeitouni 2017): writing $\beta=1+\epsilon+iR$, (i) $F_\beta=0$ for $\epsilon<0$ and for $0<\epsilon\le\epsilon_0$ with $c\sqrt{\epsilon}\le R\le c'/\sqrt{\epsilon}$ (explicit non-optimal constants); (ii) $F_\beta>0$ on the explicit curve $1+\epsilon=\frac{R}{2\pi}\log\frac{1+\pi/R}{1-\pi/R}$, $R>\pi$; (iii) a complete local description near $\beta=1$: the Fisher zeros in $\delta<|\beta-1|<c'$ lie within $C_\delta N^{-1}$ of a local critical curve $\Gamma=\{1+\epsilon\pm iR_0(\epsilon)\}$ with $|R_0(\epsilon)-\epsilon|\le C\epsilon^2$, and the scaled zero-counting measure converges weakly to an explicit density on $\Gamma$. Consistent with the physics literature (Glasser–Privman–Schulman 1986; Krasnytska et al. 2016, whose non-rigorous prediction that the zeros pinch the real axis at angle $\pi/4$ matches), these results still leave the global structure of the phase diagram undetermined: whether the two proven regions are separated by a single curve for all $R$, and what that curve is.

Problem Statement

For the Curie-Weiss model with complex inverse temperature $\beta=1+\epsilon+iR$ and zero external field, determine the global phase diagram of the free energy $F_\beta=\lim_{N\to\infty}N^{-1}\log|Z_{\beta,N}|$, where $Z_{\beta,N}$ is the Curie-Weiss partition function. Specifically, prove or refute the conjecture of Shamis and Zeitouni (2017): (a) there exists a curve $\{1+\epsilon_0(R)+iR:-\infty<R<\infty\}$ with $\epsilon_0(0)=0$, $\epsilon_0(R)>0$ for $R\neq0$, and $\lim_{R\to+\infty}R^2\epsilon_0(R)=b\in(0,\infty)$, together with a function $\delta_0(R)\ge0$ vanishing only at $R=0$, such that $F_\beta=0$ for $0\le\epsilon\le\epsilon_0(R)$ and $F_\beta>0$ for $\epsilon_0(R)<\epsilon\le\epsilon_0(R)+\delta_0(R)$; and (b) the curve is given by the branch $u^*(\beta)$ of the saddle-point equation $f'_\beta(u)=\beta u-\beta\tanh(\beta u)=0$ with $u^*(1)=0$ (for $f_\beta(u)=\beta u^2/2-\log\cosh(\beta u)$), through the implicit equation $\mathrm{Re}\, f_{1+\epsilon_0(R)+iR}(u^*(1+\epsilon_0(R)+iR))=0$. A complete solution determines the exact shape of the critical curve, including the constant $b$, and thereby the global accumulation locus of the Fisher zeros $\{\beta:Z_{\beta,N}=0\}$ as $N\to\infty$.

References

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