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Journal of Physics C Solid State Physics

Dilute Ising chains with spin S

M. F. Thomsen, Michael F. Thorpe

Published Jul 30, 1983• 7 citations
Publisher

Abstract

Randomly diluted one-dimensional Ising chains of spin S are studied by breaking the systems down to noninteracting finite segments. Exact closed-form expressions are obtained for the specific heat and perpendicular susceptibility of S= 1 / 2 chains and numerically exact results are given for spins up to 5 / 2 .
DOI
10.1088/0022-3719/16/21/020

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An exact result for the partition function for a general one-dimensional Ising chain of N spin-1/2 particles described by the Hamiltonian HN=−∑i=1NJiσiσi+1−∑i=1NHiσi is given. For an open strand, JN = 0; for a closed chain, σN+1 = σ1, JN ≠ 0. The novelty of the trick used enables one to obtain the partition function and all the spin correlation functions for open and closed chains with equal ease. Special cases of this model have been discussed before to elucidate certain features of some biological systems. New expressions for parallel and perpendicular susceptibilities for this model are also derived. When Ji and Hi are treated as random variables, the above Hamiltonian describes a one-dimensional Ising spin glass. In this case some simple models and formal averaging procedures are discussed.
Exact, closed expressions are obtained for the perpendicular and parallel susceptibilities of the one-dimensional spin-1/2 Ising model (i) with nearest neighbor interactions in a uniform magnetic field and (ii) with next-nearest neighbor interactions but without the magnetic field, in the thermodynamic limit. Graphs of the susceptibilities as a function of temperature for various magnetic field strengths in (i), and as a function of different ratios of the interaction strengths in (ii), are discussed.
Investigates the bicritical dynamics of one-dimensional bond-diluted kinetic Ising models in the percolation limit at criticality. A wide class of single-spin-flip dynamics is applied and domain boundary diffusion arguments are used to identify the dominant processes producing flipping of the ordered finite chain segments constituting the diluted chain. Exact expressions for the rate of flip as a function of length of segment are obtained and are used to derive corresponding expressions for the decay of equilibrium correlations, and of the magnetisation (from a non-equilibrium uniformly magnetised initial state) in the chain. The results are in agreement with those of earlier rigorous calculations restricted to the special case of Glauber dynamics.
The susceptibility in arbitrary directions, the specific heat, the energy and the magnetization of general Ising-type chains are exactly expressed in terms of the eigenvectors of the transfer matrix corresponding to the Ising-type interaction. These quantities, as a result, can be explicitly obtained for small spin values and generally obtained from the solution of corresponding eigenvalue problem of finite degree. Numerical estimations are easy for arbitrary spin values. This formula includes the spin- S transverse Ising model with vanishing or non-vanishing parallel external field, the Blume-Emery-Griffiths model and the Blume-Capel model, mixed spin and mixed bond periodic systems such as alternating Ising chains or Ising ferrimagnets and also includes Ising models with random structures. Low temperature behaviors of these systems and the crossover to infinite spin systems are also investigated.