Continuum Limits and Exact Finite-Size-Scaling Functions for One-Dimensional $O(N)$-Invariant Spin Models

TL;DR

This paper provides an exact solution for one-dimensional $O(N)$ spin models, explores their continuum limits, classifies universality classes, and computes finite-size-scaling functions, including special cases and new mathematical formulas.

Contribution

It offers the first exact solution for finite-size scaling in 1D $O(N)$ models and classifies their universality classes for various interactions.

Findings

Multiple universality classes identified for different interaction parameters.

Exact finite-size-scaling functions computed for these classes.

No new universality classes found in the special two-parameter family.

Abstract

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 discussed 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 finite-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated, and no additional universality classes appear. In the appendices we give new formulae 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.