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

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

2003-06-13

arxiv.org/abs/cond-mat/0306354v2

Physics

Condensed Matter

Statistical Mechanics

Short comment on possible generalization to wider classes of systems added; accepted for publication in Journal of Statistical

Scientific paper

10.1023/B:JOSS.0000022373.63640.

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.

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