A Thouless-Like Effect in the Dyson Hierarchical Model with Continuous Symmetry

TL;DR

This paper proves a Dyson conjecture relating phase transition conditions to the convergence of a series in a hierarchical ferromagnetic model, and discovers a Thouless-like effect in the distribution of normalized spins at criticality.

Contribution

It establishes the necessary and sufficient condition for phase transition in Dyson's hierarchical model with continuous symmetry, and reveals a Thouless-like phenomenon at the critical point.

Findings

Convergence of the series l_1 + l_2 + ... is necessary and sufficient for phase transition.

Spontaneous magnetization vanishes at the critical point.

Distribution of normalized average spin tends to uniform on the sphere at T_c.

Abstract

We study Dyson's classical $r$-component ferromagnetic hierarchical model with a long range interaction potential $U(i,j)= -l(d(i,j)) d^{-2}(i,j)$, where $d(i,j)$ denotes the hierarchical distance. We prove a conjecture of Dyson, which states that the convergence of the series $l_1+l_2+...$, where $l_n=l(2^n)$, is a necessary and sufficient condition of the existence of phase transition in the model under consideration, and the spontaneous magnetization vanishes at the critical point, i.e. there is no Thouless' effect. We find however that the distribution of the normalized average spin at the critical temperature $T_c$ tends to the uniform distribution on the unit sphere in $\Bbb R^r$ as the volume tends to infinity, a phenomenon which resembles the Thouless effect.