  We apply a combination of numerical renormalization group (NRG) and
renormalized perturbation theory (RPT) to a model of two quantum dots
(impurities) described by two Anderson impurity models hybridized to their
respective baths. The dots are coupled via a direct interaction $U_{12}$ and an
exchange interaction $J$. The model has two types of quantum critical points,
one at $J=J_c$ to a local singlet state and one at $U_{12}=U_{12}^c$ to a
locally charge ordered state. The renormalized parameters which determine the
low energy behavior are calculated from the NRG. The results confirm the values
predicted from the RPT on the approach to the critical points, which can be
expressed in terms of a single energy scale $T^*$ in all cases. This includes
cases without particle-hole symmetry, and cases with asymmetry between the
dots, where there is also a transition at $J=J_c$. The results give a
comprehensive quantitative picture of the behavior of the model in the low
energy Fermi liquid regimes, and some of the conclusions regarding the
emergence of a single energy scale may apply to a more general class of quantum
critical points, such as those observed in some heavy fermion systems.
