Long-range order (LRO) and phase transition in quantum anisotropic ferromagnetic ising - Heisenberg linear chain

The problem of the existence (or absence) of the long-range order (LRO) in the systems of interacting magnetic moments with various spatial dimensions D and various spin symmetry group G was of interest for a very long period and looks sometimes as being solved but, in fact, full and exact formulation still lacks. In recent years the interest arose to the magnetic systems with "low" (D<3) dimensions, because these systems are actual for possible applications, e.g., in spintronics (D=2), quantum computation (D=1) a.o.; new possibilities are now available in fabrication of the magnetic chains, layers and "spin networks". The problem of the existence of the LRO in the anisotropic quantum Heisenberg model on the D=1 uniformly periodic chain is reconsidered in view of the possibility of sufficiently slow decaying positive exchange interaction with infinite effective radius. It is shown that the macrosopic arguments given by Landau and Lifshitz and then supported microscopically by Mermin and Wagner on the base of Bogoliubov's inequalities fail for this case so that the non-zero LRO yet may exist.This result was anticipated by Thouless on the grounds of phenomenological analysis, and we give its microscopic foundation what amounts to the generalization of Mermin-Wagner theorem on the case of the infinite second moment of the exchange interaction. Some well known in lattice statistics models - i.e., Kac-I, Kac-II and Dyson, or hierarchical, model - illustrate our results; in fact, we c0nsider the equivalent Montroll-Shlesinger which allows to use the effective analytic possibilities of the probability theory. In these models the chain is considered as non-uniform, i.e. logarithmically periodic one; this causes the crossover between diffusion (or magnetic) phases with and without LRO while the order parameter instead of usual one is also logarithmic. © 2013 Nova Science Publishers, Inc. All rights reserved.