ext{C}}$ is well localized within the cluster. In reality it is difficult to find a quantity which is short ranged in the entire parameter range of the interaction strength and the doping concentration. Therefore we need to choose an appropriate quantity to periodize, according to situations. For example, $Q=\S$ ($\S$: self-energy) is a good choice for weakly interacting normal metals,[@ks01] but it becomes highly nonlocal and long ranged in the strong coupling region (e.g., in the Mott insulator). This is due to an appearance of zeros of $G$, i.e., poles of $\S$ in the momentum space (see APPENDIX A for further detail). On the other hand, in the Mott insulator $Q=G$ is more appropriate because it is nearly local in the strong coupling regime.[@kk06]

Another choice for the periodization is the cumulant $$M=[i\w_n+\mu-\S]^{-1}.$$ It was pointed out that the cumulant periodization works well in a wide range of $U$ including the strong coupling, because it is similar in the functional form to the atomic Green function.[@sk06] Moreover, the periodization by the cumulant has an important feature, i.e., it can describe both poles and zeros of $G$ at the same time, while the periodization by using $\S$ ($G$) describes only poles (zeros). This opens up the intriguing possibility of exploring the coexistence of poles and zeros at the Fermi level, as substantiation of anomalous metals.

In the present study, we adopt the cumulant periodization scheme, $Q=M$, to investigate [*metals in the vicinity of the Mott insulator*]{}. In fact, $M$ is highly local even in the doped metallic states. In APPENDIX B we demonstrate this local nature by CDMFT calculations for $N_{\text{c}}=4\times4$ cluster; we find that $M^{\text{C}}$ is nearly localized already within the inner $2\times 2$ cluster for a parameter region relevant to the present study. After obtaining the lattice cumulant $M^{\text{L}}$ through Eq. (\[eq:periodize\]), we calculate the self-energy $\S^{\text{L}}$, Green’s function $G^{\text{L}}$, and spectral function $A