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The single-orbital Anderson impurity model using graphene as the host material is considered for the case where the impurity is placed on top of a carbon atom of the graphene lattice. This is an excellent setup for the  pseudogap impurity model, where there exists a quantum phase transition from the free local moment phase to the Kondo screening phase. In this work, the scaling behavior of the spin-spin correlator at quantum critical points is numerically investigated. It shows signatures of the logarithmic correction to scaling to the lowest temperature in use. Furthermore, the result suggests that the scaling dimension might vanish as , thus the widely-accepted scaling behavior for  might be destroyed at , signifying that  is the upper critical dimension for the class of pseudogap impurity problem.

In an arbitrary system subjected to a quench, or an external field that varies the system parameters, the degrees of freedom increases double in comparison to that of an isolated system. In this study, we consider the quantum impurity system subjected to a quench, and measure the corresponding time-evolution of the spectral function, which is originated from the time-resolved photoemission spectroscopy. Due to the large number of degrees of freedom, the expression of the time-dependent spectral function is twice much more complicated than that of the time-independent spectral function, and therefore the calculation is extremely time consuming. In this paper, we estimate the scale of time consumption of such calculation in comparison to that of time-independent calculation, and present our solution to the problem by using parallel computing as implementing both MPI and OpenMP to the calculation. We also discuss the possibility to exploit parallel computing with GPU in the near future, and the preliminary results of time-dependent spectral function.