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Acta Physica Polonica A

Ising Model on Tangled Chain. II. Magnetization and Susceptibility

Rexhep Mejdani

Published Jun 1, 1994• 0 citations
Publisher

Abstract

In the preceding paper we have considered an Ising model defined on tangled chain to study the behaviour of the free energy and entropy, particul ar l y i n t he zer o-f i el d and zer o-t emper at ur e l i mi t .I n t hi s paper , f ol l owi ng the main line and basing on some results of the previous work, we shall study in the "language" of state configurations the behaviour of the magnetization and the susceptibility for different conditions of the model, to understand better the competition between the ferromagnetic bonds along the chain and the antiferromagnetic additional bonds across the chain.Particularly interesting is the behaviour of the susceptibility in the zero-field and zero-temperature limit.Exact solutions for the magnetization and susceptibility, generated by analytical calculations and iterative algorithms, are described.The additional bonds, introduced as a form of perfect disorder, indicate a particular effect on the spin correlation.We found that the condition J = -J' between the ferromagnetic interaction J along the chain and the antiferromagnetic interaction J' across the chain is somewhat as a "transition-region" condition for this behaviour.

DOI
10.12693/aphyspola.85.919

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In the preceding paper we have considered an Ising model defined on tangled chain to study the behaviour of the free energy and entropy, particul ar l y i n t he zer o-f i el d and zer o-t emper at ur e l i mi t .I n t hi s paper , f ol l owi ng the main line and basing on some results of the previous work, we shall study in the "language" of state configurations the behaviour of the magnetization and the susceptibility for different conditions of the model, to understand better the competition between the ferromagnetic bonds along the chain and the antiferromagnetic additional bonds across the chain.Particularly interesting is the behaviour of the susceptibility in the zero-field and zero-temperature limit.Exact solutions for the magnetization and susceptibility, generated by analytical calculations and iterative algorithms, are described.The additional bonds, introduced as a form of perfect disorder, indicate a particular effect on the spin correlation.We found that the condition J = -J' between the ferromagnetic interaction J along the chain and the antiferromagnetic interaction J' across the chain is somewhat as a "transition-region" condition for this behaviour.
In the preceding paper we have considered an Ising model defined on tangled cliain to study the behaviour of the free energy and entropy, particularly in the zero-field and zerotemperature limit. In this paper, following the main line and basing on some results of the previous work, we shall study in the language of state configurations the behaviour of the magnetization and the susceptibility for different conditions of the model, to understand better the competition between the ferromagnetic bonds along the chain iuid the antifcrromagnetic additional bonds across the chain. Particularly interesting is the behaviour of the susceptibility in the zero-field and zero-temperature limit. Exact solutions for the magnetization and susceptibility, generated by analytical calculations and iterative algorithms, are described. The additional bonds, introduced as a form of perfectly disorder, indicate a particular effect on the spin correlation. We found thnt the condition J — -J' between the ferromagnetic interaction J along the chain and the antiferromagnetic interaction J' across the chain is somewhat as a transition-region condition for this behaviour. MIRAMARE TRIESTE May 1993 Classification numbers: 05.50.+q,65.50+m,75.10.Hk,75.10.Nr Permanent address: Department of Physics, University of Tirana, Tirana, Albania.
In this paper we have considered an Ising model defined on tangled chain, in which more bonds have been added to those of pure Ising chain.To understand their competition, particularly between ferromagnetic and antiferromagnetic bonds, we have studied, using the transfer matrix method, some simple analytical calculations and an iterative algorithm, the behaviour of the free energy and entropy, particularly in the zero-field and zero-temperature limit, for different configurations of the ferromagnetic tangled chain and different types of additional interactions (ferromagnetic or antiferromagnetic).We found that the condition J = -J' between the ferromagnetic interaction J along the chain and the antiferromagnetic interaction J a l o n g the chain is somewhat as a "transition region" condition for this behaviour.Our results indicate also the existence of nonzero entropy at zero temperature.
The methods of Williams,Doman and Williams,and Doman and Habeeb are used to describe the ground state properties of infinite ferromagnetic Ising chains in a random field.Analytic results for the ground state energy,the magnetisation per spin in the direction of the local field and the entropy as a function of magnetic field are obtained. When the probability factor of the zero local field is zero,our results are in agreement with thut obtained previously by Williams. terminators.A terminator could be defined as the simplest group of spins whose direction is indepen- dent of that of its neighbours,but depends on the value of the magnetic field.The behaviour of the system is now dominated by these subchains and so we take into account all possible configurations for such subchains.The idea of superspins and superb- onds introduced by Williams(1981)and applied later to examine the properties of many Ising
Abstract In this paper we consider ladders with two, three, and four coupled Ising spin chains, characterized by interchain and intrachain couplings, to study in detail their magnetic behaviour for different ratios of the interaction constants and different values of magnetic field, by using a transfer matrix method and a complex algorithm, realizing the diagonalization and the differentiation. We have proposed a combined algorithm to realize also the generation of the elements of the transfer matrix for more complicated ladder configurations. The problem of Ising ladders is straightforward with a crossover between a 2D high‐temperature region and a 1D low‐temperature region when the 2D correlation length equals the width. To emphasize the fact that there is no finite magnetization in absence of the magnetic field, we study also analytically this case for the ladder with two coupled Ising chains. In presence of the magnetic field, for very weak magnetic fields (of the order 10−8, in reduced units) the magnetization goes from the value one to the value zero in a narrow interval of nonzero temperatures, where also the susceptibility exhibits a very high peak. This narrow interval, increasing the number of chains, is displaced towards higher temperatures. Also, in the case of interchain antiferromagnetic couplings, the formation of interchain spin pairs becomes clear.