Superfluid Density of Strongly Underdoped Cuprate Superconductors from a Four-Dimensional XY Model

A new phenomenology is proposed for the superfluid density ρ_s of strongly underdoped cuprate superconductors based on recent data for ultraclean single crystals of YBa₂Cu₃O_7-x. We show that the puzzling departure from Uemura scaling and the decline of the slope as the T_c=0 quantum critical point is approached can be understood in terms of the renormalization of quasiparticle effective charge by quantum fluctuations of the superconducting phase. We then employ a (3+1)-dimensional XY model to calculate, within particular approximations, the renormalization of ρ_s and its slope, explain the new phenomenology, and predict its eventual demise close to the quantum critical point.