Magnetic clustering at a quantum critical point: A modified percolation theory

The formation and dynamics of magnetic clusters have proven to be important for materials that have been driven to a quantum critical point via substantial chemical substitution. Tiny variations in the local exchange interaction lead to the formation of a distribution of Kondo temperatures, which in turn gives rise to a fragmentation of the magnetic lattice. Importantly, the temperature scale at which the clusters form is typically low enough that finite-size effects force the clusters to order internally as soon as they form. We argue that this process defies description by a standard percolation model but that a slight restriction-- whereby ordered clusters are not allowed to be broken up-- succeeds, but with the consequence that a new universality class emerges. We demonstrate this model with analytics as well as Monte-Carlo numerical results.