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

A new phenomenology is proposed for the superfluid density ${\rho }_s$ of strongly underdoped cuprate superconductors based on recent data for ultraclean single crystals of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$. 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 ${\rho }_s$ and its slope, explain the new phenomenology, and predict its eventual demise close to the quantum critical point.

Figure 1

Behavior expected in the vicinity of the $(3+z)\mathrm{D}\mathrm{\text{−}}XY$ quantum critical point in the doping-temperature plane. Solid line $T_c(x)$ represents the superconducting phase transition; dashed lines are crossovers.

Figure 2

Panel (a) shows the SCHA result, the critical theory Eq. (14), and the interpolation discussed in the text (dashed line) for the case $d=3$ with short range interactions. Inset: The solid line is the exact solution of Eq. (11a), and the dashed line is its approximation Eq. (11b). Panel (b) displays ${\rho }_s(x,T)$ from Eq. (11b) for several values of $S$. The inset illustrates the effect of anisotropy: The curves (bottom to top) correspond to $\eta =0.0,0.3,0.6,1.0$. Solid lines are solutions of Eq. (11a) for $d=2,3$ and short range interaction (we find $S/U\approx 0.040$ for $d=3$ and $S/U\approx 0.058$ for $d=2$). Dashed lines are numerical solutions of the corresponding anisotropic equations.