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European Journal of Physics

Discontinuous magnetisation in mean-field theory

Manuel Berrondo, S. Y. Larsen

Published Jul 1, 1989• 2 citations
Publisher

Abstract

The authors calculate the classical partition function for a mean-field effective Hamiltonian with a finite number of spins. They find a single-valued magnetisation as a function of the external magnetic field, with a sharp transition at zero magnetic field, below the Curie point. The authors then calculate the thermodynamic limit for the same system (equivalent to the infinite-dimensional Ising model) in a straightforward fashion, and find the corresponding discontinuous behaviour at the phase transition point, with no van der Waals loop.

DOI
10.1088/0143-0807/10/3/009

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The authors calculate the classical partition function for a mean-field effective Hamiltonian with a finite number of spins. They find a single-valued magnetisation as a function of the external magnetic field, with a sharp transition at zero magnetic field, below the Curie point. The authors then calculate the thermodynamic limit for the same system (equivalent to the infinite-dimensional Ising model) in a straightforward fashion, and find the corresponding discontinuous behaviour at the phase transition point, with no van der Waals loop.
This chapter explores the basic notions of phase transitions, and the manner in which non-analytic thermodynamic behaviour may occur in principle. It presents some concrete calculations of phase transition behaviour in magnetic systems. The chapter is concerned with the transfer matrix method, first introduced by Hendrik Kramers, which reduces the problem of calculating the partition function to the problem of finding the eigenvalues of a certain matrix. It describes the low temperature expansion and Weiss’ mean field theory for magnets. Mean field theory is the simplest treatment of an interacting statistical mechanical system, apart from the approximation of ignoring the interactions all together. The chapter provides a simple, non-systematic derivation of mean field theory for the Ising model. This mean field theory can be easily improved by applying it to clusters of spins rather than individual spins.
Making use of the Stratonovich linearization procedure for bilinear exponents and certain eigenvalue properties of the generalised Jacobi matrices, an exact expression has been obtained, in a closed form, for the partition function of the d (=3) dimensional, nearest neighbour coupled, Ising model for finite continuous spins(=D) in the presence of an external field. A Gaussian measure has been chosen for integration over the spin space. The method admits complete generalisation with respect to the dimensionality, the range of interaction and the sign of the interaction constant J (ferro- or antiferromagnetic). All thermodynamical quantities of interest can be obtained by taking suitable partial derivatives of the partition function. The critical indices can be ascertained for the cases of d=3 for which the theory predicts a second-order phase transition. For the ferromagnetic case(J>0), the spontaneous magnetisation vanishes identically the temperature T>T (the transition temperarture) and the magnetic susceptibility follows essentially the Curie Weiss Law.
Mean-field theory is a widely used approximation for describing phase transitions, particularly effective above the upper critical dimension. Its origins can be traced back to the van der Waals theory of the liquid-gas transition and Weiss's molecular field theory of the paramagnetic-ferromagnetic transition. However, it was Lev D. Landau who provided a unifying and general framework applicable to a broad class of physical systems. The mean-field approach typically involves neglecting thermal fluctuations, which is a reasonable assumption in many classical contexts. However, its application to quantum phase transitions at zero temperature is less common. The aim of this short pedagogical paper is to explore and clarify the use of the mean-field approach in the less familiar domain of quantum phase transitions. We specifically consider the Ising model and the Blume-Capel model.
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices (4×44\times 44×4 and 4×64\times 64×6) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.
In this set of notes, a complete, pedagogical tutorial for applying mean field theory to the two-dimensional Ising model is presented. Beginning with the motivation and basis for mean field theory, we formally derive the Bogoliubov inequality and discuss mean field theory itself. We proceed with the use of mean field theory to determine a magnetisation function, and the results of the derivation are interpreted graphically, physically, and mathematically. We give a new interpretation of the self-consistency condition in terms of intersecting surfaces and constrained solution sets. We also include some more general comments on the thermodynamics of the phase transition. We end by evaluating symmetry considerations in magnetisation, and some more subtle features of the Ising model. Together, a self-contained overview of the mean field Ising model is given, with some novel presentation of important results.