On the mean-field spherical model

TL;DR

This paper provides exact solutions for the mean-field spherical model, analyzing its thermodynamic properties, phase transitions, and ensemble equivalence, with insights into the topology of state space submanifolds.

Contribution

It offers the first exact solutions for the mean-field spherical model in both ensembles and explores the topological aspects related to ensemble nonequivalence.

Findings

Exact solutions for finite and infinite N in both ensembles

Identification of nonanalytic microcanonical entropy at finite N

Insights into the topology of state space submanifolds

Abstract

Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N. The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor sigma-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.