Fat-tailed and compact random-field Ising models on cubic lattices

TL;DR

This study explores the ferromagnetic Ising model on cubic lattices with non-Gaussian random fields characterized by fat-tailed or compact distributions, revealing persistent ferromagnetic order across various tail behaviors.

Contribution

Introduces a unified functional form for non-Gaussian fields in the Ising model, analyzing effects of different tail parameters on magnetic ordering in three dimensions.

Findings

Ferromagnetic order persists at finite temperatures for all studied tail parameters.

Mean-field predictions remain valid in three-dimensional cases.

Different distribution tails influence the nature of magnetic phase transitions.

Abstract

Using a single functional form which is able to represent several different classes of statistical distributions, we introduce a preliminary study of the ferromagnetic Ising model on the cubic lattices under the influence of non-Gaussian local external magnetic field. Specifically, depending on the value of the tail parameter, $\tau $ ($\tau < 3$), we assign a quenched random field that can be platykurtic (sub-Gaussian) or leptokurtic (fat-tailed) form. For $\tau< 5/3$, such distributions have finite standard deviation and they are either the Student-$t$ ($1< \tau< 5/3$) or the $r$-distribution ($\tau< 1$) extended to all plausible real degrees of freedom with the Gaussian being retrieved in the limit $\tau \rightarrow 1$. Otherwise, the distribution has got the same asymptotic power-law behaviour as the $\alpha$-stable L\'{e}vy distribution with $\alpha = (3 - \tau)/(\tau - 1)$. The uniform distribution is achieved in the limit $\tau \rightarrow \infty$. Our results purport the existence of ferromagnetic order at finite temperatures for all the studied values of $\tau$ with some mean-field predictions surviving in the three-dimensional case.