We have derived exact results for the initial static susceptibilities of nonuniform spin-1 2 Ising chains.The results obtained permit one to study the random{bond Ising chain.The in uence of several types of disorder on the temperature dependence of the initial static susceptibilities is discussed.
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.
The susceptibilities of both the one-dimensional random Ising chain and the random Ising Bethe lattice composed of plural kinds of magnetic atoms of S = 1/2 and non-magnetic atoms are studied based on a new method, i.e., by using simultaneous linear equation of order n, where n is the number of kinds of magnetic atoms. The susceptibilities of the binary and ternary systems, which correspond to n = 1 and 2, respectively, are exactly obtained for both the one-dimensional chain and the Bethe lattice. The critical temperature of the random Bethe lattice is also obtained. It is shown that the critical temperature of the binary Bethe lattice is in agreement with that of the ordinary two- or three-dimensional lattice in the cluster variation method and that the critical temperature of the ternary Bethe lattice is independent of the signs of exchange integrals. It is also conjectured that the critical temperature of the ordinary ternary system does not depend on the signs of exchange integrals when two kinds of magnetic atoms are of the same type and the lattice structure is loose-packed. The problem of the random exchange integrals is also discussed.
We have carried out Monte Carlo studies of a random-exchange antiferromagnetic spin-(1/2) chain. For systems with XY-like (anisotropic) and with Heisenberg (isotropic) coupling, our results confirm the existence of a disorder-induced low-temperature (T) divergence in the long-wavelength Sz-Sz susceptibility \ensuremath{\chi} which was previously predicted by real-space renormalization-group (RSRG) treatments. Over the finite temperature range studied, these results are consistent with a 1/(T ln2T) behavior of \ensuremath{\chi}, and hence in qualitative agreement with the RSRG results. As in the XY-Heisenberg regime, we also find a disorder-induced enhancement of the low-T susceptibility for a system with Ising-like exchange coupling which, over the finite temperature range studied, is again consistent with RSRG results. However, there are inconsistencies between the RSRG predictions in the Ising-like regime at very low temperatures, and the exact results for the random-exchange Ising chain and the low-temperature behavior of \ensuremath{\chi} in the Ising-like regime may in fact be more complicated than predicted by RSRG. Finally, we also present results for the antiferromagnetic susceptibility and structure factor. For both Heisenberg and Ising-like systems, we find that disorder suppresses the long-range antiferromagnetic correlations at low T.