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Journal of physics. A, Proceedings of the Physical Society. General

Distribution of zeros of the partition function for the one dimensional Ising models

Seiichiro Katsura, M Ohminami

Published Jan 1, 1972• 23 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/5/1/014

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Statistical Thermodynamics of Finite Ising Model. II

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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.

Journal of the Physical Society of Japan•Oct 1, 1970•44 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.
To study the relations between Yang-Lee's theory of the phase transitions and the distribution of the zeros of the partition functions in the antiferromagnetic case, the partition functions of finite (4×4 and 4×6) Ising model with the nearest and the next nearest neighbor interactions are calculated. The distributions of the zeros in the fugacity z plane of the partition function for various combinations of the values of J / k T - J '/ k T or tanh ( J /2 k T )-tanh ( J '/2 k T ) are investigated. The patterns of the distribution of the zeros show several types corresponding to ferromagnetic state, antiferromagnetic state and superantiferromagnetic state. In the antiferromagnetic ordered state the locus of the zeros is found to be nearly two concentric circles and to cross the positive real axis at two points z c and 1/ z c when the next neighbor ferromagnetic interaction is introduced. This means that our system shows the phase transitions at finite critical fields H c and - H c . The locus of zeros of Fisher's model of super-exchange antiferromagnet is also shown to cross the positive real axis at z c and 1/ z c below the Néel temperature.
Zero distributions of the partition function for a Ising model of 6×6 square lattice with the first ( J 1 ) and the second ( J 2 ) neighbor interactions and loci of the zeros of that for 4×∞ lattice are obtained. From the results of the present and previous studies, the followings are concluded. The locus of the Ising model at low temperature is a unit circle when J 1 >0 and J 1 +2 J 2 >0; dual circles when J 1 <0 and J 2 >0, (or J 1 >0 and J 2 <0) and in the region of the first order transition; dual loops with cusps when J 1 <0, J 2 >0 and in the region of the second order transition; four-fold loops with negative real tail when J 1 <0 and J 2 >0. Loci of zeros for the hard core lattice gas of 6×∞ lattice with and without the second neighbor interaction are also obtained. The locus of the hard core lattice gas with attraction is nearly a circle below the tricritical temperature, and a loop with a cusp above the tricritical temperature. Apparent contradictions to the Yang-Lee theory observed in earlier stages disappear.
The distribution of zeros of the partition function for one-dimensional lattice system of generalized Ising spins with magnitude unity is studied, where it is assumed that the component spins interact with the nearest neighbours through a ferromagnetic coupling and each spin state has a certain degeneracy. The spin states are denoted by variable σ which can take either of three values, plus and minus unities and zero with the respective degeneracies, D(1), D(0) and D(-1), where it is assumed D(1) = D(-1). zeros of the partition function Z(ζ) of the system as exposed to an external magnetic field are studied on the ζ-plane, where ζ=exp(H) indicates the exponential function of the external field divided by temperature H and it is regarded as a complex number. The distribution of zeros is classified into three typical patterns, corresponding to the respective three regions on the plane of temperature(T) versus degneracy ratio (ρ=D(0)/D(1)): Zeros never lie on the unit circle in the region of larger ρ, all on the unit circle in the region of smaller ρ and only a part on the unit circle in the remaining region. On the basic of this result, it is expected in the case of two- or three-dimensional system that phase trasitions of first and second orders occur in tha cases of ρ larger and smaller than acritical calue ρc, respectively.
The distributions of the partition function zeros in the complex a=e2betaJ1 plane of the square-lattice Ising model with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions are investigated as a function of R=J2/J1. Starting from the well-known two-circle distribution of the zeros a=+/-1+sqrt[2]eitheta for R=0 , finally the partition function zeros lie on the unit circle a=eitheta for R=infinity . Between these two ends, the changes in the zero distributions are described. Using the partition function zeros, the critical point ac(R) and the thermal scaling exponent yt(R) are estimated for the Ising ferromagnet (equivalently, antiferromagnet) and superantiferromagnet. For the special case of R=1/2 , the possible implications of the zero distributions are also discussed.
To study the locus of zeros of the partition function for the antiferromagnets, an exactly soluble simplest model (the antiferromagnetic Husimi-Temperley model) is proposed. Thermodynamic properties of the model are shown to be the same as those in the molecular field approximation of the Ising or Heisenberg model. The locus of zeros in the complex fugacity plane of the model is obtained by the method of comparing the real part of the various branches of the complex free energies. The locus is a loop which does not cut the real positive axis above the critical temperature. Below the critical temperature, the locus of the zeros consists of dual loops which cut the real positive axis at z c and 1/z c and the density of zeros near the real axis is proportional to mod z-z c mod . At the critical temperature the locus is a loop which crosses at z=1 and the density of zeros is proportional to mod z-z c mod 3 .
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.