Superconductivity in the two-dimensional Hubbard model with cellular dynamical mean-field theory: A quantum impurity model analysis

Doping a Mott insulator gives rise to unconventional superconducting correlations. Here we address the interplay between $d$-wave superconductivity and Mott physics using the two-dimensional Hubbard model with cellular dynamical mean-field theory on a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette. Our approach is to study superconducting correlations from the perspective of a cluster quantum impurity model embedded in a self-consistent bath. At the level of the cluster, we calculate the probabilities of the possible cluster electrons configurations. Upon condensation, we find an increased probability that cluster electrons occupy a four-electron singlet configuration, enabling us to identify this type of short-range spin correlation as key to superconducting pairing. The increased probability of this four-electron singlet comes at the expense of a reduced probability of a four-electron triplet with no significant probability redistribution of fluctuations of charges. This allows us to establish that superconductivity at the level of the cluster primarily involves a reorganization of short-range spin correlations rather than charge correlations. We gain information about the bath by studying the spectral weight of the hybridization function. Upon condensation, we find a transfer of spectral weight leading to the opening of a superconducting gap. We use these insights to interpret the signatures of superconducting correlations in the density of states of the system and in the zero-frequency spin susceptibility.