We present the phase diagram of clusters made of two, three and four coupled Anderson impurities. All three clusters share qualitatively similar phase diagrams that include Kondo screened and unscreened regimes separated by almost critical crossover regions reflecting the proximity to barely avoided critical points. This suggests the emergence of universal paradigms that apply to clusters of arbitrary size. We discuss how these crossover regions of the impurity models might affect the approach

more »

... o the Mott transition within a cluster extension of dynamical mean field theory. Acknowledgments 28 Appendix. CFT at work 28 References 34 Acronyms and main notation W Non-interacting electron bandwidth U Hubbard on-site repulsion T * F Quasi-particle effective Fermi temperature Impurity hybridization width T K Kondo temperature ρ( ) Impurity density of states (iω) Impurity self-energy in Matsubara frequencies MIT Mott metal-to-insulator transition DMFT Dynamical mean field theory NRG Wilson's numerical renormalization group CFT Conformal field theory DOS Single-particle density of states Figure 3 . Behavior versus U and doping δ of the quasiparticle Fermi temperature, T * F , which translates within DMFT into the Kondo temperature T K of the effective impurity model. J is an effective intra-cluster energy scale. near a Mott transition. As we previously mentioned, these clusters may be experimentally realized on metallic surfaces or, eventually, by arranging quantum dots in proper geometries. Moreover, these models represent a theoretical challenge by themselves, which requires the full machinery of Wilson's numerical renormalization group (NRG) [27] [28] [29] and conformal field theory (CFT) [30] for a detailed comprehension. Before entering into the details of our calculations, it is worth briefly presenting the physical idea that guided this work. First of all, let us recall some basic facts of the singlesite DMFT mapping onto impurity models. Within this mapping, the quasi-particle effective Fermi temperature, T * F , translates into the Kondo temperature, T K , of the impurity model. The self-consistency condition causes T K to vanish at a finite value of U , which signals, in the lattice counterpart, the onset of the Mott transition. This also implies that the metallic phase just prior to the Mott transition translates into an Anderson impurity model deep inside the Kondo regime, with a very narrow Kondo resonance and pre-formed Hubbard side-bands [21] . The same behavior should occur even when dealing with a cluster of impurities, which should therefore translate into a cluster of Kondo impurities. The novelty stems from the other energy scales which we collectively denoted as J , and that take care of quenching in the Mott insulator the degrees of freedom other than the charge. Indeed, near the Mott transition, J translates into additional processes, like for instance a direct exchange between the impurity spins, which tend to remove, completely or partially, the degeneracy of the cluster. Consequently, J competes with the Kondo effect, an agent that takes more advantage the more degenerate the impurity cluster ground state. We notice that this competition is always active in impurity clusters, while it is commonly absent in single-impurity models except in multi-orbital cases [31, 32] whose physics is in fact close to clusters. We believe that this additional ingredient is precisely the common denominator of all impurity cluster models, which endows them with the capability of providing a more faithful description of a realistic Mott transition within DMFT. Indeed, in the presence of the intra-cluster coupling J , the approach to the Mott transition changes as qualitatively shown in figure 3 , with a Kondo temperature smoothly decreasing from its initial value W as U/W increases and becoming of order J just before the transition. Analogously, see also figure 3, starting from the Mott insulator and doping it, T K will smoothly increase from its value T K = 0 at zero doping, until it will again cross a value of order J . In other words, any impurity cluster should experience, within DMFT, two different regimes. The first, when T K J , in which full Kondo screening takes place and the impurity density of states displays the usual Kondo resonance. In the lattice model, this regime translates into a conventional correlated metal. The second, when T K J , particularly close to the Mott transition, in which no or only partial Kondo screening occurs. Here the impurity density