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International Journal of Quantum Chemistry

Spin‐free quantum chemistry. XIV. The infinite interaction range model for ferromagnetism

F. A. Matsen, J.G. Cosgrove, J. Michael Picone

Published Nov 1, 1973• 4 citations
Publisher

Abstract

Abstract The infinite interaction range model (IIRM) for ferromagnetic systems is presented in its spin‐free formulation. In this formulation the states are labelled by partitions which provide a natural variable for thermodynamic computation. We have extended the calculations of Kittel and Shore by computing to a practical thermodynamic limit (N ∼ 100,000). The heat capacity, magnetic susceptibility and the magnetization of the first two functions exhibit a critical temperature while the magnetization is zero at zero field for all temperatures. Spontaneous magnetization is obtained by linear extrapolation from high field or equivalently by a polarized partition function. Relationships are explored among IIRM, the Heisenberg model and the mean field model. Application to IIRM of the Yang‐Lee condition for a phase transition yields a critical temperature identical to that obtained by the direct calculation.

DOI
10.1002/qua.560070605

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Abstract The infinite interaction range model (IIRM) for ferromagnetic systems is presented in its spin‐free formulation. In this formulation the states are labelled by partitions which provide a natural variable for thermodynamic computation. We have extended the calculations of Kittel and Shore by computing to a practical thermodynamic limit (N ∼ 100,000). The heat capacity, magnetic susceptibility and the magnetization of the first two functions exhibit a critical temperature while the magnetization is zero at zero field for all temperatures. Spontaneous magnetization is obtained by linear extrapolation from high field or equivalently by a polarized partition function. Relationships are explored among IIRM, the Heisenberg model and the mean field model. Application to IIRM of the Yang‐Lee condition for a phase transition yields a critical temperature identical to that obtained by the direct calculation.
The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.
We investigate low- and high-temperature properties of the ferromagnetic Ising model of infinite-spin. By using the cummulant expansion methods, we obtain low-temperature expansions of the free energy, the specific heat, the spontaneous magnetization, the zero-field susceptibility and so on, and high-temperature expansions of the free energy and the zero-field susceptibility. We compare the results with those obtained by Thompson for the system on the one-dimensional lattice, with those by using Monte Carlo simulations for the systems on the square lattice and on the simple cubic lattice and with those with the aid of the mean-field approximation.
The Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.
The thermodynamic quantities of finite two- and three-dimensional ferromagnetic X Y -model such as 3×3 and 2×2×2 lattices have been calculated for the purpose of investigating the properties of liquid Helium II. The density dependence of the λ-line for the finite quantum lattice gas model agrees with the results of molecular field approximation and the experimental data of pure He 4 liquid near the λ-transition. The possibility of the occurrence of negative thermal expansion coefficient below the λ-point was roughly discussed with the use of the same consideration as the Ising model of Lee and Yang. The magnetization versus field curve tends to a step function when the temperature becomes smaller than the critical temperature estimated from the maximum of the specific heat. There exists a great difference between the pure isotropic Heisenberg model and X Y -model in the case of ferro-magnetic interaction at zero-temperature.
A long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.
The Lee-Yang theorem for the zeros of the partition function of a ferromagnetic Ising model with real pair spin interactions is extended to general Ising models with complex many-spin interactions (satisfying appropriate ``ferromagnetic'' and spin inversion symmetry conditions). When many-spin interactions are present, all zeros lie on the imaginary Hz-axis for sufficiently low (but fixed) T, but, in general, some leave the imaginary axis as T → ∞. The extended Ising theorem is used to prove the same result for a Heisenberg system of arbitrary spin with the real anisotropic pair interaction Hamiltonian Hij=−(JijzSizSjz+JijxSixSjx+JijySiySjy)in an arbitrary transverse field (Hx, Hy) under the ``ferromagnetic'' condition Jijz≥|Jijx| and Jijz≥|Jijy|. The analyticity of the limiting free energy of such a Heisenberg ferromagnet and the absence of a phase transition are thereby established for all (real) nonzero magnetic fields Hz. The Ising theorem is also applied to hydrogen-bonded ferroelectric models to prove, in particular, that the zeros for the KDP model lie on the imaginary electric field axis for all T below the transition temperature Tc.