Critical Properties of an Ising Model with Dilute Long-range Interactions

TL;DR

This paper investigates the critical behavior of a diluted long-range Ising model with random neighbors, revealing a phase transition with mean field exponents and a logarithmic length scale for finite connectivity.

Contribution

It introduces a model of Ising spins with random long-range interactions and analyzes its critical properties, bridging local and mean field behaviors.

Findings

Second order phase transition with mean field exponents

Critical length scale scales as log of system size

Transition persists for finite connectivity greater than 2

Abstract

Statistical mechanical models with local interactions in $d>1$ dimension can be regarded as $d=1$ dimensional models with regular long range interactions. In this paper we study the critical properties of Ising models having $V$ sites, each having $z$ randomly chosen neighbors. For $z=2$ the model reduces to the $d=1$ Ising model. For $z= \infty$ we get a mean field model. We find that for finite $z > 2$ the system has a second order phase transition characterized by a length scale $L={\rm ln}V$ and mean field critical exponents that are independent of $z$.