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Journal of Physics A Mathematical and General

The frustrated quantum spin-1/2 chain

P Reed

Published Nov 21, 1992• 1 citation
Publisher

Abstract

The Trotter decomposition and transfer matrix methods have been used to study the frustrated quantum spin-1/2 chain. The random bond model in a uniform field and ferromagnetic models in a random field have been investigated. The results indicate the existence of non-zero entropy at zero temperature. The random bond model also exhibits non-analytic behaviour of the magnetization as a function of applied field.

DOI
10.1088/0305-4470/25/22/014

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The Trotter decomposition and transfer matrix methods have been used to study the frustrated quantum spin-1/2 chain. The random bond model in a uniform field and ferromagnetic models in a random field have been investigated. The results indicate the existence of non-zero entropy at zero temperature. The random bond model also exhibits non-analytic behaviour of the magnetization as a function of applied field.
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.
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.
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.
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.