Rigorous Analysis of Singularities and Absence of Analytic Continuation at First Order Phase Transition Points in Lattice Spin Models

TL;DR

This paper rigorously proves that thermodynamic potentials in certain lattice spin models exhibit non-analytic behavior at first order phase transition points, with a detailed analysis of the role of interaction range and phase droplet stability.

Contribution

It provides two new rigorous results on the non-analytic properties of thermodynamic potentials at first order phase transitions in lattice models and Kac potentials.

Findings

Pressure has no analytic continuation at the transition point for models with finite ground states.

Non-analytic behavior relates to the interaction range in Kac potential models.

A crossover exists between non-analytic finite-range models and analytic mean field models.

Abstract

We report about two new rigorous results on the non-analytic properties of thermodynamic potentials at first order phase transition. The first one is valid for lattice models ($d\geq 2$) with arbitrary finite state space, and finite-range interactions which have two ground states. Under the only assumption that the Peierls Condition is satisfied for the ground states and that the temperature is sufficiently low, we prove that the pressure has no analytic continuation at the first order phase transition point. The second result concerns Ising spins with Kac potentials $J_\gamma(x)=\gamma^d\phi(\gamma x)$, where $0<\gamma<1$ is a small scaling parameter, and $\phi$ a fixed finite range potential. In this framework, we relate the non-analytic behaviour of the pressure at the transition point to the range of interaction, which equals $\gamma^{-1}$. Our analysis exhibits a crossover between the non-analytic behaviour of finite range models ($\gamma>0$) and analyticity in the mean field limit ($\gamma\searrow 0$). In general, the basic mechanism responsible for the appearance of a singularity blocking the analytic continuation is that arbitrarily large droplets of the other phase become stable at the transition point.