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

Distribution of zeros of the partition function of the antiferromagnetic Husimi-Temperley model I

M Ohminami, Yoshihiko Abe, Seiichiro Katsura

Published Dec 1, 1972• 3 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/5/12/006

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Physical Review Letters•Sep 23, 2004•68 citations
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 .
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.
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.
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.
From the exact partition functions of the Ising model on L x L square lattices (up to L = 14) in an arbitrary nonzero external magnetic field H, the precise distributions of the Fisher zeros in the complex temperature (a = e(2betaJ)) plane are obtained. Our results indicate that in the limit L-->infinity the Fisher zeros for H not = 0 do cut the positive real axis at the antiferromagnetic critical point ac(H); these values are compared with the results of closed-form approximations for the antiferromagnetic critical line. From the Fisher zeros the thermal scaling exponent yt = 1 is also obtained along the critical line, indicating the logarithmic singularity of the specific heat for the antiferromagnetic Ising model even in a strong uniform magnetic field.
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.
There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).