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Journal of the Physical Society of Japan

Statistical Mechanics of the Finite Heisenberg Model. II

Chikao Kawabata

Published Apr 1, 1970• 11 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.28.861

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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 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 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.
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 exact excitation spectrum of a 3×3 two dimensional square lattice interacting via the nearest neighbour anisotropic Heisenberg Hamiltonian H = J ∑ < i j > { S i z S j z +γ( S i x S j x + S i y S j y )}, has been obtained for various values of the anisotropy constant γ. The zeros of the partition function in the complex µ=exp (- m H e / k B T ) plane are calculated for different γ for both ferromagnetic ( J <0) and antiferromagnetic ( J >0) coupling. For 0⩽γ⩽10{\leqslant}\gamma{\leqslant}10⩽γ⩽1, the ferromagnetic zeros obey the generalized Lee-Yang theorem at all temperatures, but, for γ>1, they violate it at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the (unphysical) negative real axis for all values of γ and temperature. The zero-field susceptibility, for antiferromagnetic coupling, does not display a local maximum as a function of temperature in contrast to Kawabata's results on the same lattice.
The partition function of a three-dimensional Ising model was calculated by a high-speed computer for a cubic array of twenty-seven points. The periodicity condition was imposed. The specific heat of the 3×3×3 lattice was calculated as a function of temperature. The critical temperature estimated from the specific heat curve seems slightly lower that obtained from the Padé approximation. The zeros of the partition function of the complex temperature plane were also obtained. The critical temperature estimated from the zeros near the positive real axis seems slightly higher than both of the above values. Similar calculations were carried out for the case of the two-dimensional Ising model. Several results are compared with those obtained from Onsager's exact solution.