General non-existence theorem for phase transitions in one-dimensional systems with short range interactions, and physical examples of such transitions

TL;DR

This paper critically reviews the limitations of van Hove's theorem on phase transitions in one-dimensional short-range systems, presents examples of such transitions, and proves a broader non-existence theorem with implications for physical models.

Contribution

It introduces a generalized non-existence theorem for phase transitions in one-dimensional short-range systems, extending beyond van Hove's original scope.

Findings

Van Hove's theorem has limited applicability.

Examples of one-dimensional models with phase transitions are provided.

A new, broader non-existence theorem is proven.

Abstract

We examine critically the issue of phase transitions in one-dimensional systems with short range interactions. We begin by reviewing in detail the most famous non-existence result, namely van Hove's theorem, emphasizing its hypothesis and subsequently its limited range of applicability. To further underscore this point, we present several examples of one-dimensional short ranged models that exhibit true, thermodynamic phase transitions, with increasing level of complexity and closeness to reality. Thus having made clear the necessity for a result broader than van Hove's theorem, we set out to prove such a general non-existence theorem, widening largely the class of models known to be free of phase transitions. The theorem is presented from a rigorous mathematical point of view although examples of the framework corresponding to usual physical systems are given along the way. We close the paper with a discussion in more physical terms of the implications of this non-existence theorem.