Effects of on-site Gaussian disorder in the ferromagnetic Blume Capel model

We conduct a numerical study of a generalization of the Blume-Capel model in which the crystal field $D_i$, which controls the local vacancy density, is site dependent rather than uniform. We employ Markov Chain Monte Carlo to compute the shifted phase transition line $T_c$ as a function of the width $\sigma$ of a gaussian distribution $P(D_i)\sim \exp (-(D_i-\bar{D}{)}^2\mathrm{/}2{\sigma }^2)$ at fixed $\bar{D}$. We characterize how the presence of this on-site Gaussian disorder reduces $T_c$ across the second order transition line, and discuss how our results suggest that $T_c$ remains finite even at large values of $\sigma$. We also determine the effects of disorder on the tricritical point of the conventional, uniform, Blume-Capel model, finding that it significantly suppresses the first-order phase transition, eliminating it entirely at large values of $\sigma$.