Upper critical dimension in a quantum impurity model: Critical theory of the asymmetric pseudogap Kondo problem

Impurity moments coupled to fermions with a pseudogap density of states display a quantum phase transition between a screened and a free moment phase upon variation of the Kondo coupling. We describe the universal theory of this transition for the experimentally relevant case of particle-hole asymmetry. The theory takes the form of a crossing between effective singlet and doublet levels, interacting with low-energy fermions. Depending on the pseudogap exponent, this interaction is either relevant or irrelevant under renormalization group transformations, establishing the existence of an upper-critical ``dimension'' in this impurity problem. Using perturbative renormalization group techniques we compute various critical properties and compare with numerical results.