Magnons in the Heisenberg model with a dynamical scaling symmetry

A special Heisenberg model is considered for which the exchange integral takes on the same value J not only for geometrically equivalent neighbours, but also for such j-neighbours which constitute an orbit of a “hidden” symmetry group of scaling transformations. The dispersion law for magnons for this model constitutes a reproduction of some rules of arithmetic number theory. The extra symmetry is illustrated by a chain of 12 spins, which may be shown to be equivalent to a toroidal 4×3-periodic crystal.