The mean-field phi4-model: entropy, analyticity, and configuration space topology

TL;DR

This paper uses large deviation techniques to analyze the mean-field phi4-model, revealing that phase transition nonanalyticities arise from maximization over magnetization, not from entropy nonanalyticity, challenging the topological hypothesis.

Contribution

It provides an exact expression for configurational entropy and explains the origin of phase transition nonanalyticities without topology changes, highlighting a mechanism specific to long-range interactions.

Findings

Entropy s(v,m) is analytic in both variables.

Nonanalyticity in s(v) arises from maximization over m.

Phase transition does not necessarily involve topology change.

Abstract

A large deviation technique is applied to the mean-field phi4-model, providing an exact expression for the configurational entropy s(v,m) as a function of the potential energy v and the magnetization m. Although a continuous phase transition occurs at some critical energy v_c, the entropy is found to be a real analytic function in both arguments, and it is only the maximization over m which gives rise to a nonanalyticity in s(v)=sup_m s(v,m). This mechanism of nonanalyticity-generation by maximization over one variable of a real analytic function is restricted to systems with long-range interactions and has--for continuous phase transitions--the generic occurrence of classical critical exponents as an immediate consequence. Furthermore, this mechanism can provide an explanation why, contradictory to the so-called topological hypothesis, the phase transition in the mean-field phi4-model need not be accompanied by a topology change in the family of constant-energy submanifolds.