Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Unknown eBooks

Ising model on tangled chain. Part 2: Magnetization and susceptibility

Rexhep Mejdani

Published May 1, 1993• 1 citation
OpenAlex

Abstract

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.

Related papers

Ising Model on Tangled Chain. II. Magnetization and Susceptibility

Rexhep Mejdani

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.

Acta Physica Polonica A•Jun 1, 1994•0 citations

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

Rexhep Mejdani

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.

Acta Physica Polonica A•Jun 1, 1994•3 citations

Correlation function of random-field Ising chains: is it Lorentzian or not?

J. M. Luck, Theo M. Nieuwenhuizen

The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.

Journal of Physics A Mathematical and General•Jun 21, 1989•18 citations

Ground State Properties of Frustrated Ferromagnetic Ising Chains in a Uniform-Random-External Magnetic Field.

Dr.MANSOOR Dr.MANSOOR, A-Z. HABEEB

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

Journal of Engineering Research (Libya)•Jun 30, 1995•0 citations

The exact solution of magnetic susceptibility for finite Ising ring with a magnetic impurity

Da-cheng Ma, Yan Qi, An Du

We connected the two ends of a finite spin-1/2 antiferromagnetic Ising chain with a magnetic impurity at one end to form a closed ring, and studied its magnetic susceptibility exactly by using the transfer matrix method. We calculated the magnetic susceptibility in the whole temperature range and gave the phase diagram at ground state of the system about the anisotropy of the impurity and strength of the connection exchange interaction for spin-1 and -3/2 impurities. We also gave the ground state entropy of the system and derived the asymptotic expression of the magnetic susceptibility multiplied by temperature at the zero-temperature limit and the high-temperature limit. It was found that a degenerate phase may exist in some parameter region at zero temperature for the odd spin number system, and the ground state entropy is ln(2) in the nondegenerate phase and is dependent on the spin number in the degenerate phase. The magnetic susceptibility of the system at low temperature exhibits ferromagnetic behavior, and the Curie constant is related to the spin configuration at ground state. When the ground state is nondegenerate, the Curie constant is equal to the square of the net spin, regardless of the parity of the number of the spin. When the spin number is odd and the ground state is degenerate, the Curie constant may be related to the total spin number. At the high-temperature limit, the magnetic susceptibility multiplied by temperature is related to the spin quantum number of the impurity and the spin number in the ring.

Canadian Journal of Physics•Jun 29, 2021•5 citations

Critical behavior of random transverse-field Ising spin chains

Daniel S. Fisher

A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.

Physical review. B, Condensed matter•Mar 1, 1995•828 citations
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 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 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 two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.
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
We connected the two ends of a finite spin-1/2 antiferromagnetic Ising chain with a magnetic impurity at one end to form a closed ring, and studied its magnetic susceptibility exactly by using the transfer matrix method. We calculated the magnetic susceptibility in the whole temperature range and gave the phase diagram at ground state of the system about the anisotropy of the impurity and strength of the connection exchange interaction for spin-1 and -3/2 impurities. We also gave the ground state entropy of the system and derived the asymptotic expression of the magnetic susceptibility multiplied by temperature at the zero-temperature limit and the high-temperature limit. It was found that a degenerate phase may exist in some parameter region at zero temperature for the odd spin number system, and the ground state entropy is ln(2) in the nondegenerate phase and is dependent on the spin number in the degenerate phase. The magnetic susceptibility of the system at low temperature exhibits ferromagnetic behavior, and the Curie constant is related to the spin configuration at ground state. When the ground state is nondegenerate, the Curie constant is equal to the square of the net spin, regardless of the parity of the number of the spin. When the spin number is odd and the ground state is degenerate, the Curie constant may be related to the total spin number. At the high-temperature limit, the magnetic susceptibility multiplied by temperature is related to the spin quantum number of the impurity and the spin number in the ring.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.