Despite more than two decades of intensive investigations,
the true nature of high temperature (high-Tc) superconductivity
observed in the cuprates remains 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 of the superconducting
order parameter, has gained significant support from various experiments.
However, the microscopic mechanism responsible for the surprisingly 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 the superconductivity literally
into a well-understood physics of superfluid in a special dilute bosonic
system of local pairs. In the prototypical material (La1?xSrx)2CuO4,
without the use of any free parameter, a d-wave superconductivity is obtained
for doping above 5.2%, below which unexpected incoherent p-wave pairs dominate.
Throughout the whole underdoped region, very soft phases are found to originate
from enormous mass enhancement of the pairs. Furthermore, a striking mass divergence
is predicted that dictates the occurrence of the observed quantum critical point.
Finally, good theoretical agreement with experiments will be presented to address
several other unexplained superfluid behaviors.