Monte Carlo simulations of magnetic clustering at a quantum critical point*

We present the results of Monte Carlo simulations on a percolating magnetic system with relevance to quantum critical point materials. It has previously been shown that, for heavily doped quantum critical point compounds such as Ce(Ru$_{0.24}$Fe$_{0.76})_{2}$Ge$_{2}$, the formation and dynamics of magnetic clusters strongly influences the physical response of the system at low temperature. Our simulation is based on the idea that finite-size effects force small magnetic clusters to order at comparatively high temperatures and, once formed, are impervious to Kondo shielding. Disorder acts to introduce a distribution of Kondo temperatures which, in turn, governs the formation of clusters as the temperature is lowered. We implement a percolation model based on such a distribution-- first introduced by Bernal et al.-- and with a restriction whereby Kondo shielding is allowed to remove moments from the infinite cluster \textit{only}. We investigate how this influences thermodynamic quantities as well as how well the simulations align with our analytic theory that is based on the same restriction.