Weak phase stiffess and nature of the quantum critical point in underdoped cuprates*

We demonstrate that the zero-temperature superconducting phase diagram of underdoped cuprates can be quantitatively understood in the strong binding limit, using only the experimental spectral function of the ``normal'' pseudo-gap phase without any free parameter. In the prototypical (La$_{1-x}$Sr$_x$)$_2$CuO$_4$, a kinetics-driven $d$-wave superconductivity is obtained above the critical doping $\delta_c\sim 5.2\%$, below which complete loss of superfluidity results from local quantum fluctuation involving local $p$-wave pairs. Near the critical doping, a enormous mass enhancement of the local pairs is found responsible for the observed rapid decrease of phase stiffness. Finally, a striking mass divergence is predicted at $\delta_c$ that dictates the occurrence of the observed quantum critical point and the sudden suppression of the Nernst effects in the nearby region.