  We address the phenomenon of critical Kondo destruction in pseudogap
Bose-Fermi Anderson and Kondo quantum impurity models. These models describe a
localized level coupled both to a fermionic bath having a density of states
that vanishes like |\epsilon|^r at the Fermi energy (\epsilon=0) and, via one
component of the impurity spin, to a bosonic bath having a sub-Ohmic spectral
density proportional to |\omega|^s. Each bath is capable by itself of
suppressing the Kondo effect at a continuous quantum phase transition. We study
the interplay between these two mechanisms for Kondo destruction using
continuous-time quantum Monte Carlo for the pseudogap Bose-Fermi Anderson model
with 0<r<1/2 and 1/2<s<1, and applying the numerical renormalization-group to
the corresponding Kondo model. At particle-hole symmetry, the models exhibit a
quantum critical point between a Kondo (fermionic strong-coupling) phase and a
localized (Kondo-destroyed) phase. The two solution methods, which are in good
agreement in their domain of overlap, provide access to the many-body spectrum,
as well as to correlation functions including, in particular, the
single-particle Green's function and the static and dynamical local spin
susceptibilities. The quantum-critical regime exhibits the hyperscaling of
critical exponents and \omega/T scaling in the dynamics that characterize an
interacting critical point. The (r,s) plane can be divided into three regions:
one each in which the calculated critical properties are dominated by the
bosonic bath alone or by the fermionic bath alone, and between these two
regions, a third in which the bosonic bath governs the critical spin response
but both baths influence the renormalization-group flow near the quantum
critical point.
