Superfluid stiffness, Mott transition and competition between antiferromagnetism and superconductivity in cuprates

Superfluid stiffness $\rho_s$ allows a superconductor to establish phase coherence and to sustain a supercurrent. When $\rho_s$ is small, phase coherence may occur at a lower temperature than Cooper pair formation, lowering the critical temperature $T_c$ below its mean-field value $T_{\text{MF}}$. This occurs because of phase fluctuations. Coexistence of $d$-wave superconductivity with other phases in underdoped cuprates, such as antiferromagnetism (AF) or charge-density waves (CDW), may enhance the phase fluctuations and hence lower $T_c$. To shed light on this physics, the zero-temperature value of $\rho_s=\rho_{zz}$ along the $c$-axis was computed for different values of Hubbard interaction $U$ and different sets of tight-binding parameters describing the high-temperature superconductors YBCO and NCCO. We used Cellular Dynamical Mean-Field Theory for the one-band Hubbard model with exact diagonalization as impurity solver and state-of-the-art bath parametrization. We conclude that Mott physics plays a dominant role in determining the superfluid stiffness on the hole-doped side of the phase diagram while on the electron-doped side it is competition between antiferromagnetism and d-wave superconductivity that plays a dominant role in determining the value of $\rho_{zz}$ near half-filling: Antiferromagnetism wins over superconductivity near half-filling while near optimal doping on the underdoped side, homogeneous coexistence between superconductivity and antiferromagnetism causes the superfluid stiffness to drop sharply. This may account for the lowering of $T_c$ just below optimal doping in electron-underdoped cuprates. At large overdoping, $\rho_{zz}$ behaves in a more BCS-like manner in both the electron- and hole-doped cases.