The generalization of the Mermin–Wagner theorem and the possibility of long-range order in the isotropic discrete one-dimensional quantum Heisenberg model

The problem of existence of long-range order in the isotropic quantum Heisenberg model on the D=1 lattice is reconsidered in view of the possibility of sufficiently slow decaying 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 fail for this case so that the non-zero spontaneous magnetization may yet exist. This result was anticipated by Thouless on the grounds of phenomenological analysis, and we give its microscopic foundation, which amounts to the generalization of Mermin–Wagner theorem for the case of the infinite second moment of the exchange interaction. Two well known in lattice statistics models – i.e., Kac-I and Kac-II – illustrate our results.