Numerical simulations of random spin (and fermionic) models with a wide distribution of energy scales

The magnetic behavior of semiconductors doped with randomly distributed magnetic elements (such as iron or manganese) and/or bound carriers (such as phosphorus or boron in silicon) are described by many-body Hamiltonians with a broad distribution of coupling constants and energy scales. These wide distributions (covering several orders of magnitude in some cases) lead to unusual properties, such as strong suppression of magnetic phase transitions due to quantum fluctuations, unusual thermodynamic behavior in the magnetically ordered phase, etc. The wide distributions also pose several challenges to both analytical and computational approaches used to calculate the physical properties of such systems. We describe some of the techniques that have been applied successfully to such systems, including numerical renormalization group as well as Monte Carlo methods. Examples are drawn from lightly doped conventional semiconductors [Si, Ge] as well as diluted magnetic semiconductors [such as (Cd, Mn)Te and (Ga, Mn)As]. Extension of these methods to diluted magnetic semiconductors in the metallic regime with itinerant carriers (fermionic degrees of freedom) is also discussed.