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Journal of the Physical Society of Japan

Statistical Mechanics of the Finite Heisenberg Model. III

Chikao Kawabata

Published Jun 1, 1970• 8 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.28.1396

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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.
The eigenfunctions of a two-dimensional finite 3×3 lattice of Heisenberg model ( s =1/2) are obtained exactly with the aid of group theoretical technique and by using a computer. The eigenvalues of the anisotropic exchange Hamiltonian have been computed. The zeros of the partition function for the two-dimensional finite 3×3 lattice lie on the same unit circle in the complex magnetic field plane as those obtained in the three-dimensional finite 2×2×2 lattice. The zeros of the partition function obtained by coarse-graining of eigenvalues distribute on a line in the complex temperature plane. The thermodynamic quantities–specific heat, entropy, magnetization–has been numerically computed for ferro- and antiferromagnetic coupling. The critical index of the magnetization versus field is discussed on the basis of these numerical results.
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.
Experimental properties of the lambda line of liquid helium are used together with properties of the zero-field spin 1/2 XY model to determine, on the one hand, the point on the lambda line to which the zero-field model applies, and on the other hand, to determine the lattice spacing of the model. On the f.c.c. lattice the specific heat of the model, C/NkB ~ −0.255 ln (1 − Tc/T) − 0.190 near Tc. This result is shown to be in good agreement with recent experimental results of Ahlers on liquid helium. Other properties of the S = 1/2 XY model are also calculated and, where comparison can be made, agree well with experiment.
The eigen-functions and eigen-values of finite three-dimensional Heisenberg models with anisotropy constant as a parameter (spin=1/2) were obtained exactly with the aid of group theoretical technique and by using a high speed computer. The calculated eigenvalues are identical with those obtained by Serber and Dresselhaus for the isotropic simple cubic 2×2×2 lattice. However, it was found that some terms in their full expressions were to be corrected. The zeros of the partition function of the above 2×2×2 lattice have been investigated as functions of the anisotropy parameter both in the complex temperature plane and in the complex magnetic field plane. Thermodynamic functions such as energy, specific heat, magnetization, and susceptibility have been numerically computed for these finite system.
The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.
A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.