Ising model with spins S=1/2 and 1 on directed and undirected Erd\"os-R\'enyi random graphs

TL;DR

This study uses Monte Carlo simulations to analyze the phase transitions of the Ising model with spins 1/2 and 1 on directed and undirected Erd"os-Rényi graphs, revealing mean-field behavior and first-order transitions depending on parameters.

Contribution

It provides new insights into the phase transition behavior of the Ising model on ER graphs with different spins and directions, highlighting the conditions for spontaneous magnetization and phase transition order.

Findings

Mean-field spontaneous magnetization at p=z/N for spin 1/2

No spontaneous magnetization at percolation threshold p=1/N

First-order phase transition for spin 1 at z=4 and 9 neighbors

Abstract

Using Monte Carlo simulations we study the Ising model with spin S=1/2 and 1 on {\it directed} and {\it undirected} Erd\"os-R\'enyi (ER) random graphs, with $z$ neighbors for each spin. In the case with spin S=1/2, the {\it undirected} and {\it directed} ER graphs present a spontaneous magnetization in the universality class of mean field theory, where in both {\it directed} and {\it undirected} ER graphs the model presents a spontaneous magnetization at $p = z/N$ ($z=2, 3, ...,N$), but no spontaneous magnetization at $p = 1/N$ which is the percolation threshold. For both {\it directed} and {\it undirected} ER graphs with spin S=1 we find a first-order phase transition for z=4 and 9 neighbors.