Topology, phase transitions and the spherical model

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.

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Cited by in corpus (11)

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• Exploring the energy landscape of XY models
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• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Nonanalyticities of entropy functions of finite and infinite systems
• Topological Theory of Phase Transitions
• Geometrical aspects in the analysis of microcanonical phase-transitions
• Phase transitions in simplified models with long-range interactions
• Phase transitions and topology in 2+k XY mean-field models
• Thermo-Statistical description of the Hamiltonian non extensive systems: The reparametrization invariance