Phase diagram of a pseudogap Anderson model with application to graphene

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.