Long-range interactions and non-extensivity in ferromagnetic spin models

TL;DR

This paper investigates long-range ferromagnetic spin models with decaying interactions, proposing a generalized Curie-Weiss model to analyze non-extensive regimes and supporting the conjecture that mean field theory is exact in these cases.

Contribution

It introduces a generalized Curie-Weiss model for non-extensive regimes and supports the conjecture that mean field theory is exact across all interaction decay rates.

Findings

Monte Carlo simulations support the conjecture for d=1.

A scaling law unifies extensive and non-extensive regimes.

The model extends understanding of long-range interactions in spin systems.

Abstract

The Ising model with ferromagnetic interactions that decay as $1/r^\alpha$ is analyzed in the non-extensive regime $0\leq\alpha\leq d$, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the $N\rightarrow\infty$ limit ($N$ being the number of spins) we propose a generalization of the Curie-Weiss model, for which the $N\rightarrow\infty$ limit is well defined for all $\alpha\geq 0$. We conjecture that mean field theory is {\it exact} in the last model for all $0\leq\alpha\leq d$. This conjecture is supported by Monte Carlo heat bath simulations in the $d=1$ case. Moreover, we confirm a recently conjectured scaling (Tsallis\cite{Tsallis}) which allows for a unification of extensive ($\alpha>d$) and non-extensive ($0\leq\alpha\leq d$) regimes.