Scaling in Condensed Matter Physics: the Kondo Problem, Aggregation, and the Random-Field Ising Model.

Scaling concepts are applied to three problems in condensed matter physics. The N -orbital, single impurity Kondo problem is shown to be equivalent to an N -state Potts model in one dimension with inverse-square interactions. Using renormalization techniques, it is found that the peak in the static magnetic susceptibility at finite N develops into a discontinuity as Ntoinfty. A scaling hypothesis is consistent with an increase in the peak height as ln N. The practice of using a high -energy cutoff such as the bandwidth as an effective temperature is discussed. Secondly, a generalized model of aggregation is introduced in which particles interact at a distance. For infinite-range interactions, the model is solved exactly. A scaling ansatz, similar to that made in finite-size scaling of critical phenomena, is used to extrapolate the results to finite range. Computer simulations support this ansatz. Lastly, the two-dimensional random-field Ising model is used to model monolayer adsorption onto a substrate with quenched, random impurities. From renormalization -group arguments, we obtain expressions for the zero-field susceptibility as a function of the linear dimension of a typical crystallite and the width of the field distribution. The Curie-law divergence of the susceptibility found at low temperatures for pure finite-sized crystallites is removed in the presence of impurities.