Superfluid density in cuprate high-$T_c$ superconductors: A new paradigm

We show that the Uemura relation, namely $T_c$ proportional to superfluid density ${\rho }_s,$ is not generally satisfied in high-$T_c$ superconductors. Instead, a modified plot of $T_c/{\Delta }_0$ versus ${\rho }_s$ where ${\Delta }_0$ is the maximum d-wave gap at $T=0,$ exhibits universal features that point to an alternative interpretation of the underlying physics. In the underdoped region this plot exhibits the canonical negative curvature expected when a ground-state correlation competes with superconductivity (SC) by opening up a gap (the pseudogap) in the normal-state density of states. Moreover we observe simple linear relations between $T_c{\rho }_s,$ the electronic entropy at $T_c,$ and the critical impurity concentration for suppressing SC. These relations confirm that the pseudogap persists to $T=0,$ strongly modifying the SC ground state.