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Journal of Kufa for Mathematics and Computer

Ising Chains Properties at Low Temperature Extended Analysis

Mansoor A.Z. Habeeb, Nazera Khalil Dakhil, Fatimah O. Salman

Published Dec 1, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.31642/jokmc/2018/020206

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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.
Recently published methods used to examine the ground-state properties of the random-bond Ising chain in a uniform magnetic field are extended to describe the system at low temperatures. The entropy spikes broaden asymmetrically, and the excitation energy and specific heat become asymmetric double-peaked functions of the magnetic field with zero values between the peaks at B=2J/r for integer r. The magnetisation steps at B=2J/r are smoothed out asymmetrically. Both the ground-state and finite-temperature properties of a simple periodic frustrated Ising chain are investigated using the transfer-matrix method. The ground-state properties are similar to those of the random bond chain; the ground-state entropy, for instance, is finite for J<or=B<or=2J, with spikes at J and 2J.
For pt.II see ibid., vol.15, p.1693 (1982). The methods developed by Williams (1981) to describe the ground-state properties of a random-bond infinite Ising chain are extended to apply to a system of finite-length random-bond Ising chains. It is shown how to evaluate the magnetisation, ground-state energy, and entropy as a function of magnetic field. It is shown that the steps in the magnetisation and entropy when B=2J/r behave differently for r even and r odd. The magnetisation and entropy remain finite as B becomes very small.
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
A magnetic property of the one dimensional spin 5/2 Ising model under the magnetic field has been investigated by means of transfer matrix method. Thermodynamic response functions are also obtained for varying values of scaling temperature and scaling magnetic field. Entropy and heat capacity of the system were calculated by benefiting from the temperature dependencies of Helmholtz free energy. Our simulation results demonstrate that as the strength of the magnetic field is increased, heat capacity tend to shift to the relatively higher temperature regions, and these findings are consistent with previous results for the low spin values in one dimensional Ising systems.
The low-temperature behavior ($T\ensuremath{\ll}h, J$) of a random-bond Ising chain in a magnetic field is considered [$H=\ensuremath{-}(\frac{1}{2})J\ensuremath{\Sigma}{T}_{i}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{i+1}\ensuremath{-}h\ensuremath{\Sigma}{\ensuremath{\sigma}}_{i}, {\ensuremath{\sigma}}_{i}=\ifmmode\pm\else\textpm\fi{}1$, Ti{{T}_{i}}Ti​ is a fixed random sequence of numbers +1 and -1 with concentrations c1{c}_{1}c1​ and ${c}_{2}=1\ensuremath{-}{c}_{1}$, respectively]. The ground-state energy E0{E}_{0}E0​, magnetization ${\ensuremath{\mu}}_{0}$ and zero-point entropy S0{S}_{0}S0​ are calculated exactly. It is shown that ${\ensuremath{\mu}}_{0}$ and S0{S}_{0}S0​ are discontinuous functions of magnetic field having jumps at h=Jnh=\frac{J}{n}h=nJ​, n=1,2,…n=1, 2, \dots{}n=1,2,….
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.