CONTINUUM LIMITS AND EXACT FINITE-SIZE-SCALING FUNCTIONS FOR ONE-DIMENSIONAL O(N)-INVARIANT SPIN MODELS

We solve exactly the general one-dimensional O(N)-invariant spin model taking values in the sphere S-N-1, with nearest-neighbor interactions , in finite volume with periodic boundary conditions, by an expansion in hyperspherical harmonics. The possible continuum limits are discuss ed for a general one-parameter family of interactions and an infinite number of universality classes is found. For these classes we compute the finite-size-scaling functions and the leading corrections to finit e-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated and no a dditional universality classes appear. In the appendices we give new f ormulae for the Clebsch-Gordan coefficients and 6-j symbols of the O(N ) group, and some new generalizations of the Poisson summation formula ; these may be of independent interest.