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Physical Review Letters

Variational Approximation to the Ising Model in a Magnetic Field

S. Krinsky, Craig A. Tracy, Martin Blume

Published Apr 16, 1973• 4 citations
Publisher

Abstract

A variational approximation to the Ising model in a magnetic field ($T<~{T}_{c}$), 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.

DOI
10.1103/physrevlett.30.750

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Journal de Physique I•Mar 1, 1991•4 citations
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.
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.
Previous series expansion studies of the magnetization and susceptibility of the Ising model are extended to the higher derivatives of the free energy with respect to the magnetic field. It is found that just above the critical temperature the 2 n th derivative is given by where in two dimensions γ = 1¾, Δ = 1fraction seven-eighths and in three dimensions γ = 1¼, Δ = 1fraction nine-sixteens. In two dimensions γ is exact and Δ is correct to 0.6%; the three-dimensional indices are correct to 0.2%. Just below the critical temperature where in three dimensions 0.303 <or= β <or= 0.318 and 1·55 <or= Δprime <or= 1.65. If the even derivatives are to have the same index above and below T c , then β = fraction five-sixteens and Δprime = 1fraction nine-sixteens. The latter index lies very close to the lower limit so that, although we cannot rule out Δprime = Δ, Δprime = 1fraction five-eighths is not unlikely. In two dimensions our results are consistent with linearity of the index and, if this is assumed, then β = fraction one-eighths and Δprime = 1fraction seven-eighths exactly and there is symmetry about T c . The amplitudes C 2 n + and C n - are estimated for n = 1 to 6.