  The Anderson model of an $s$-wave single-orbital correlated impurity placed
on a noninteracting honeycomb lattice, a simplified model for studying an
impurity on graphene, is used to investigate pseudogap Kondo problem. In this
model, there are two quantum phases: the phase of free impurity local moment
and the Kondo phase where this local moment is fully screened. The transition
between these two phases is under investigation. The work focuses mostly on the
case where the impurity is placed on top of a lattice site. In this case, the
full phase diagram is constructed using three parameters: the Hubbard
interaction $U$, the hybridization strength $v_0$ and the impurity energy level
$\epsilon_d$. The phase diagram exhibits linear $(U^c, \epsilon_d^c)$ phase
boundary, the slope of which, as well as the critical value $\epsilon_d^c$,
depends strongly on $v_0^2$. Further analysis shows that the real part of the
self energy at zero frequency and the impurity occupancy can help to understand
the behaviors of the phase boundaries. The dependence of the phase transition
on the impurity position is briefly discussed, revealing difficulties that one
needs to solve in order to realize the pseudogap Kondo model in the realistic
graphene lattice.

Strongly Correlated Electrons