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Despite more than two decades of intensive investigations, the true nature ofhigh temperature (high-$T_c$) superconductivity observed in the cupratesremains elusive to the researchers. In particular, in the so-called`underdoped' region, the overall behavior of superconductivity deviates$qualitatively$ from the standard theoretical description pioneered by Bardeen,Cooper and Schrieffer (BCS). Recently, the importance of phase fluctuation ofthe superconducting order parameter has gained significant support from variousexperiments. However, the microscopic mechanism responsible for thesurprisingly soft phase remains one of the most important unsolved puzzles.Here, opposite to the standard BCS starting point, we propose a simple,solvable low-energy model in the strong coupling limit, which maps thesuperconductivity literally into a well-understood physics of superfluid in aspecial dilute bosonic system of local pairs of doped holes. In theprototypical material (La$_{1-\delta}$Sr$_\delta$)$_2$CuO$_4$, without use ofany free parameter, a $d$-wave superconductivity is obtained for doping above$\sim 5.2\%$, below which unexpected incoherent $p$-wave pairs dominate.Throughout the whole underdoped region, very soft phases are found to originatefrom enormous mass enhancement of the pairs. Furthermore, a striking massdivergence is predicted that dictates the occurrence of the observed quantumcritical point. Our model produces properties of the superfluid in goodagreement with the experiments, and provides new insights into several currentpuzzles. Owing to its simplicity, this model offers a paradigm of great valuein answering the long-standing challenges in underdoped cuprates.

Weak phase stiffness and mass divergence of superfluid in underdoped cuprates