  The numerical renormalization group method is used to investigate zero
temperature phase transitions in quantum impurity systems, in particular in the
particle-hole symmetric soft-gap Anderson model. The model displays two stable
phases whose fixed points can be built up of non-interacting single-particle
states. In contrast, the quantum phase transitions turn out to be described by
interacting fixed points, and their excitations cannot be described in terms of
free particles. We show that the structure of the many-body spectrum of these
critical fixed points can be understood using renormalized perturbation theory
close to certain values of the bath exponents which play the role of critical
dimensions. Contact is made with perturbative renormalization group
calculations for the soft-gap Anderson and Kondo models. A complete description
of the quantum critical many-particle spectra is achieved using suitable
marginal operators; technically this can be understood as epsilon-expansion for
full many-body spectra.
