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Physica A Statistical Mechanics and its Applications

Partition function zeros for the two-dimensional Ising model II

John Stephenson

Published May 1, 1986• 17 citations
Publisher
DOI
10.1016/0378-4371(86)90047-6

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The loci of zeros of the partition functions of the following one dimensional Ising models are obtained, (i) Those of S = 1 / 2 and with the nearest (J) and the next nearest (J') neighbour interactions. The loci of zeros are classified in several patterns according to a combination of J and J'. In particular when J<0 and J'>0, the existence of two nearly concentric circular arcs whose heads approach points on the positive real axis of z corresponding to the critical field at zero temperature is confirmed. This suggests that the locus of the two and three dimensional Ising antiferromagnet (J<0, J'>0) will be closed two concentric circles below the Neel temperature. (ii) Those of S=1 and of S= 3 / 2 with the nearest neighbour (J) interaction. The locus shrinks to 25 points ( theta = 2 pi j/(2S+1), J=1,2,...,2S) on the unit circle at T = infinity . It extends along the unit circle for J>0 and along a nearly radial direction for J<0 when the temperature decreases.

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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.

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Partition function of the three-dimensional Ising model

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Theoretical and Mathematical Physics•Sep 1, 1978•20 citations
The loci of zeros of the partition functions of the following one dimensional Ising models are obtained, (i) Those of S = 1 / 2 and with the nearest (J) and the next nearest (J') neighbour interactions. The loci of zeros are classified in several patterns according to a combination of J and J'. In particular when J<0 and J'>0, the existence of two nearly concentric circular arcs whose heads approach points on the positive real axis of z corresponding to the critical field at zero temperature is confirmed. This suggests that the locus of the two and three dimensional Ising antiferromagnet (J<0, J'>0) will be closed two concentric circles below the Neel temperature. (ii) Those of S=1 and of S= 3 / 2 with the nearest neighbour (J) interaction. The locus shrinks to 25 points ( theta = 2 pi j/(2S+1), J=1,2,...,2S) on the unit circle at T = infinity . It extends along the unit circle for J>0 and along a nearly radial direction for J<0 when the temperature decreases.
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.