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Journal of Contemporary Physics (Armenian Academy of Sciences)

Fisher zeros on the Husimi lattice

H. A. Lazaryan

Published Feb 26, 2011• 3 citations
Publisher

Abstract

Fisher zeros for the partition function with respect to a temperature-dependent parameter are studied. The Ising approximation for Heisenberg model with two- and three-site exchange interactions on the Husimi lattice was used. This model approximates the third layer of 3 He, absorbed on the surface of graphite (kagome lattice). Using dynamic approach, we have found an exact recursion relation for the partition function. The presence of a phase transition, both in the real and complex regions on the temperature plane was shown.

DOI
10.3103/s1068337211020058

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Fisher zeros for the partition function with respect to a temperature-dependent parameter are studied. The Ising approximation for Heisenberg model with two- and three-site exchange interactions on the Husimi lattice was used. This model approximates the third layer of 3 He, absorbed on the surface of graphite (kagome lattice). Using dynamic approach, we have found an exact recursion relation for the partition function. The presence of a phase transition, both in the real and complex regions on the temperature plane was shown.
The Ising approximation of the Heisenberg model in a strong magnetic field, with two-, and three-spin exchange interactions are studied on a Husimi lattice. This model can be considered as an approximation of the third layer of 3He absorbed on the surface of graphite (kagome lattice). Using dynamic approach we have found exact recursion relation for the partition function. For different values of exchange parameters and temperature the diagrams of magnetization are plotted and showed that magnetization properties of the model vary from ferromagnetic to antiferromagnetic depending from the value of model parameters. For antiferromagnetic case magnetization plateau at 1/3 of saturation field is obtained. Lyapunov exponent for recursion relation are considered and showed absents of bifurcation points in thermodynamic limit. The Yang-Lee zeros are analyzed in terms of neutral fixed points and showed that Yang-Lee zeros of the model are located on the arcs of the circle with the radius R=1.
The Ising approximation of the Heisenberg model in a strong magnetic field, with two-, and three-spin exchange interactions are studied on a Husimi lattice. This model can be considered as an approximation of the third layer of 3 He absorbed on the surface of graphite (kagome lattice). Using dynamic approach, we have found exact recursion relation for the partition function. For different values of exchange parameters and temperature, the diagrams of magnetization are plotted and showed that magnetization properties of the model vary from ferromagnetic to antiferromagnetic depending on the value of model parameters. For antiferromagnetic case magnetization plateau at 1/3 of saturation field is obtained. Lyapunov exponent for recursion relation are considered and showed absence of bifurcation points in thermodynamic limit. The Yang–Lee zeros are analyzed in terms of neutral fixed points and showed that Yang–Lee zeros of the model are located on the arcs of the circle with the radius R = 1.
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 .
Distribution of zeros of partition function Z without magnetic field is studied for some two-dimensional Ising models with nearst-neighbor interactions. The distributions are presented graphically for the honeycomb, triangular diced and Kagomé lattices. It is shown that an asymptotic form of high temperature expansion for ln Z is closely related with the distribution of zeros. The expansion coefficients are derived up to large orders by computer for the honeycomb and Kagomé lattices. It turns out that their oscillatory behaviors are understood very well by studying the zeros off the positive real axis, in particular the period of oscillation for the Kagomé lattice is proved to be about 5.25.
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 partition function zeros of the Ising model on a kagomé lattice in the complex magnetic-field plane are obtained. The properties of the Ising model on a kagomé lattice are not known in an external magnetic field. We estimate the joint density of states for the Ising model on a kagomé lattice in an external magnetic field by using the Wang-Landau sampling method. From the joint density of states, we evaluate its partition function zeros (so-called the Yang-Lee zeros) in the complex magnetic-field plane. For the first time, we investigate the unknown properties of the Ising model on a kagomé lattice in an external magnetic field by using its Yang-Lee zeros.