Evidence of two-dimensional quantum critical behavior in the superfluid density of deeply underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x}$

Evidence of two-dimensional (2-D) quantum critical fluctuations is observed in the superfluid density n$_{s}$(T) $\propto \quad \lambda ^{-2}$(T) of deeply underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x }$(Bi-2212). Quantum critical behavior is indicated by the evolution of the T-dependence of n$_{s}$(T)/n$_{s}$(0), which loses any evidence for thermal critical behavior and becomes quasi-linear when underdoping drops the transition temperature T$_{c}$ below roughly 48K. Two-dimensionality is indicated by the linear scaling of transition temperature T$_{c}$ with n$_{s}$(0). The 2-D behavior contrasts with that of the less anisotropic YBa$_{2}$Cu$_{3}$O$_{7-\delta }$, which sustains 3D quantum critical fluctuations.