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Physical review. B, Solid state

Variational approximation to a ferromagnet in a magnetic field

S. Krinsky, Craig A. Tracy, Martin Blume

Published Jun 1, 1974• 1 citation
Publisher

Abstract

We consider a ferromagnet in an external magnetic field with $T\ensuremath{\le}{T}_{c}$. Using a variational approximation based on the zero-field solution, we obtain an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. Explicit numerical calculations have been carried out for the two-dimensional Ising model, and we find that our results provide a good approximation to the results of series expansions throughout the region $T\ensuremath{\le}{T}_{c}$. We also apply our procedure to the eight-vertex model. Metastability can be discussed within our approximation in a manner analogous to the way it is dealt with in mean-field theory. In our treatment, the critical exponents have nonclassical values, but the pseudocritical exponents all represent square-root singularities, which is the same result as found within mean-field theory.

DOI
10.1103/physrevb.9.4808

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We consider a ferromagnet in an external magnetic field with T\ensuremath≤TcT\ensuremath{\le}{T}_{c}T\ensuremath≤Tc​. Using a variational approximation based on the zero-field solution, we obtain an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. Explicit numerical calculations have been carried out for the two-dimensional Ising model, and we find that our results provide a good approximation to the results of series expansions throughout the region T\ensuremath≤TcT\ensuremath{\le}{T}_{c}T\ensuremath≤Tc​. We also apply our procedure to the eight-vertex model. Metastability can be discussed within our approximation in a manner analogous to the way it is dealt with in mean-field theory. In our treatment, the critical exponents have nonclassical values, but the pseudocritical exponents all represent square-root singularities, which is the same result as found within mean-field theory.
A variational approximation to the Ising model in a magnetic field (T< TcT<~{T}_{c}T< Tc​), based on the zero-field solution, yields an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. In two dimensions, these are numerically evaluated, and compared with the results of series expansions.
The fermionic version of the two-dimensional Ising model in a uniform magnetic field is studied. Its approximate but rather accurate solution was obtained recently in a variational procedure based on the BCS ansatz for the ground state. Here, this solution is continued analytically into the complex H-plane in the vicinity of the origin. In the ferromagnetic phase an essential singularity in the free-energy is observed at the point H=0 of the type predicted by the droplet condensation theory. Simple phenomenological estimates for the radius of the critical droplet and for the rate of decay of metastable states are confirmed by microscopic calculations.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
A method of approximate calculation of the magnetization of an Ising magnet located in an external magnetic field is proposed. This method is based on using an exact or approximate expression for the spontaneous magnetization of an Ising magnet on the same lattice. The method uses the ratio of effective fields for clusters of one and two lattice nodes. Using the proposed method for a magnet located in an external field, the dependence of magnetization on temperature and the magnitude of the external field is calculated. The proposed method is applied in the work to the solution in the mean field approximation, to the solution in the Beta approximation and to the exact solution on a square lattice; for all these solutions, critical indices characterizing the behavior of a magnet in an external field are calculated. Keywords: the Ising model, phase transitions, the influence of an external field.