Hierarchical Spherical Model as a Viscosity Limit of Corresponding O(N)Heisenberg Model

TL;DR

This paper extends classical results of the spherical and O(N) Heisenberg models with long-range hierarchical interactions, including convergence and solution properties, and explores their behavior as the hierarchical block size approaches one.

Contribution

It adapts the Kac–Thompson solution to hierarchical models lacking translation invariance and extends the convergence proof of the Heisenberg model to the spherical model for long-range interactions.

Findings

Extended the Kac–Thompson solution to hierarchical models.

Proved convergence of O(N) Heisenberg to the spherical model in hierarchical setting.

Analyzed the model's behavior as the hierarchical block size approaches one.

Abstract

The O(N) Heisenberg and spherical models with interaction given by the long range hierarchical Laplacean are investigated. Two classical results are adapted. The Kac--Thompson solution [KT] of the spherical model, which holds for spacially homogeneous interaction, is firstly extended to hierarchical model whose interaction fails to be translation invariant. Then, the convergence proof of O(N) Heisenberg to the spherical model by Kunz and Zumbach [KZ] is extended to the long range hierarchical interaction. We also examine whether these results can be carried over as the size of the hierarchical block L goes to 1. These extensions are considered a preliminary study prior the investigation of the model by renormalization group given in [MCG] where central limit theorems for the spherical (N=\infty) model on the local potential approximation (L\downarrow 1) are then established from an explicit solution of the associate nonlinear first order partial differential equation.