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Acta Physica Polonica A

Ising Model on Tangled Chain. I. Free Energy and Entropy

Rexhep Mejdani

Published Jun 1, 1994• 3 citations
Publisher

Abstract

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.

DOI
10.12693/aphyspola.85.907

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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.
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.
The transfer matrix method is used to describe the properties of the random bond of a simple periodic frustrated Ising chain in a uniform magnetic field. Doman and Williams(1982) are found expressions for the thermodynamic functions of the system at low temperature. Their methods is extended to apply to a system of five-antiferromagnetic bond Ising chain terminated by a ferromagnetic bond at low temperatures. New expressions for the magnetization, energy, and entropy as a functions of magnetic field is found. All results are in agreement with the previous results.
The Ising model of an antiferromagnet exhibits an anomalous entropy peak at a critical magnetic field which remains even at the absolute zero of temperature, in violation of the third law of thermodynamics. By investigating numerically the behaviour of finite chains of spins interacting through the more general anisotropic Heisenberg coupling, described by the Hamiltonian provided it is shown that the zero point anomalous entropy is peculiar to the Ising model (γ = 0). If the anisotropy is sufficiently great, however, an appreciable entropy peak does persist down to low temperatures although in the anisotropic (pure Heisenberg) limit the entropy has only a broad maximum which falls steadily in height as T approaches zero.
Abstract. The antiferromagnetic spin-1/2 Ising model with the presence of the multisite interaction and in the external magnetic field on the infinite tetrahedral chain is solved exactly by using the transfer matrix method. Using the exact expression for the free energy per site of the model the thermodynamic properties of the model are investigated in detail. It is shown that the total magnetisation per site forms the magnetisation plateaus at low temperatures, the properties of which depend on the strength of the multisite interaction. The set of all possible ground states is found and discussed. The existence of the so-called single-point ground states is shown and their properties are studied in detail. The entropy per site of the model is investigated and the exact expressions for the residual entropies of all ground states are found and their macroscopic degeneracies are discussed. The existence of the ground states with the same value of magnetisation but with different entropies is observed. In addition, the behaviour of the susceptibility and the specific heat is also studied.