Phase transitions in Ising model on a Euclidean network

TL;DR

This paper studies the phase transition behavior of the Ising model on a one-dimensional Euclidean network with long-range bonds, revealing a finite temperature transition across the entire range of the decay parameter and varying critical exponents.

Contribution

It demonstrates the existence of a finite temperature phase transition for all decay parameters and characterizes the change in critical exponents depending on the value of , providing new insights into Ising models on Euclidean networks.

Findings

Finite temperature phase transition exists for all  in [0, 2).

Mean field behavior for  in [0, 1).

Non-mean field critical exponents for  in (1, 2].

Abstract

A one dimensional network on which there are long range bonds at lattice distances $l>1$ with the probability $P(l) \propto l^{-\delta}$ has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbours apart from their nearest neighbours for $0 \leq \delta < 2$. It is observed that there is a finite temperature phase transition in the entire range. For $0 \leq \delta < 1$, finite size scaling behaviour of various quantities are consistent with mean field exponents while for $1\leq \delta\leq 2$, the exponents depend on $\delta$. The results are discussed in the context of earlier observations on the topology of the underlying network.