Application of the $O(N)$-Hyperspherical Harmonics to the Study of the Continuum Limits of One-Dimensional $\sigma$-Models and to the Generation of High-Temperature Expansions in Higher Dimensions

TL;DR

This paper introduces an exact solution for one-dimensional $O(N)$-invariant spin models using hyperspherical harmonics and explores their continuum limits and high-temperature expansions, with applications to higher-dimensional models.

Contribution

It presents a novel exact solution method for $O(N)$ spin models and extends the technique to higher dimensions for analyzing continuum limits and high-temperature behavior.

Findings

Exact solution for 1D $O(N)$ models using hyperspherical harmonics

Analysis of continuum limits of the models

Extension of methods to higher-dimensional models

Abstract

In this talk we present the exact solution of the most general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbour interactions, and we discuss the possible continuum limits. All these results are obtained using a high-temperature expansion in terms of hyperspherical harmonics. Applications in higher dimensions of the same technique are then discussed.