  In quantum critical heavy fermion systems, local moments are coupled to both
collective spin fluctuations and conduction electrons. As such, the Bose-Fermi
Kondo model, describing the coupling of a local moment to both a bosonic and a
fermionic bath, has been of extensive interest. For the model in the presence
of SU(2) spin rotational symmetry, questions have been raised about its phase
diagram. Here we develop a version of continuous-time Quantum Monte Carlo
(CT-QMC) method suitable for addressing this issue; this procedure can reach
sufficiently low temperatures while preserving the SU(2) symmetry. Using this
method for the Bose-Fermi Anderson model, we clarify the renormalization-group
fixed points and the phase diagram for the case with a constant fermionic-bath
density of states and a power-law bosonic-bath spectral function
$\rho_{b}(\omega) \propto \omega^{s}$ ($0<s<1$). We find two types of Kondo
destruction QCP, depending on the power-law exponent $s$ in the bosonic bath
spectrum. For $s^{*}<s<1$, both types of QCPs exist and, in the parameter
regime accessible by an analytical $\epsilon$-expansion renormalization-group
calculation (here $\epsilon=1-s$), the CT-QMC result is fully consistent with
prior predictions by the latter method. For $s<s^{*}$, there is only one type
of QCP. At both type of Kondo destruction QCPs, we find that the exponent of
the local spin susceptibility $\eta$ obeys the relation $\eta=\epsilon$, which
has important implications for Kondo destruction QCP in the Kondo lattice
problem.
