Effect of platy- and leptokurtic distributions in the random-field Ising model: Mean field approach

Abstract

The influence of the tail features of the local magnetic field probability density function (PDF) on the ferromagnetic Ising model is studied in the limit of infinite range interactions. Specifically, we assign a quenched random field whose value is in accordance with a generic distribution that bears platykurtic and leptokurtic distributions depending on a single parameter $\tau < 3$ to each site. For $\tau< 5/3$, such distributions, which are basically Student-$t$ and $r$-distribution extended for all plausible real degrees of freedom, present a finite standard deviation, if not the distribution has got the same asymptotic power-law behavior as a $\alpha $-stable L\'evy distribution with $\alpha = (3 - \tau)/(\tau - 1)$. For every value of $\tau $, at specific temperature and width of the distribution, the system undergoes a continuous phase transition. Strikingly, we impart the emergence of an inflexion point in the temperature-PDF width phase diagrams for distributions broader than the Cauchy-Lorentz ($\tau = 2$) which is accompanied with a divergent free energy per spin (at zero temperature).