Topology, phase transitions and the spherical model

TL;DR

This paper investigates the relationship between topology changes in configuration space and phase transitions in the spherical model, finding that topological discontinuities do not necessarily indicate phase transitions.

Contribution

It provides an exact characterization of the topology of the configuration space for the short-range spherical model and shows that topological changes are not sufficient to predict phase transitions.

Findings

Topology changes occur without phase transitions for d >= 3

Discontinuities in topological functions do not coincide with phase transitions

First example where topological changes are not sufficient for phase transitions

Abstract

The Topological Hypothesis states that phase transitions should be related to changes in the topology of configuration space. The necessity of such changes has already been demonstrated. We characterize exactly the topology of the configuration space of the short range Berlin-Kac spherical model, for spins lying in hypercubic lattices of dimension d. We find a continuum of changes in the topology and also a finite number of discontinuities in some topological functions. We show however that these discontinuities do not coincide with the phase transitions which happen for d >= 3, and conversely, that no topological discontinuity can be associated to them. This is the first short range, confining potential for which the existence of special topological changes are shown not to be sufficient to infer the occurrence of a phase transition.