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The relationship between the transition temperatures TC and superfluid densities nS(0) of cuprate superconductors has been a central issue in cuprate superconductivity from the beginning. When mobile holes are removed from optimally doped CuO2 planes, TC and nS(0) decrease in a surprisingly correlated fashion. Recent measurements of the superfluid density of strongly underdoped YBa2Cu3O7- films and crystals have found a square-root scaling, TC  nS(0) where  ≈ ½, which supplants the approximately linear proportionality that had been deduced long ago from less underdoped samples by Uemura et al. and had been ascribed to the quasi-2D structure of cuprates. We measured nS(T) in films of Y1−xCaxBa2Cu3O7− as thin as two CuO2 bilayers. TC’s were as low as 3 K. We observed the 2D Kosterlitz–Thouless–Berezinski drop in nS at TC, and we recovered the linear scaling TC  nS(0) expected in 2D due to fluctuations in the phase of the superconducting order parameter. Taken together, results on 3D and 2D samples suggest that the disappearance of superconductivity with underdoping is ultimately due to quantum fluctuations near a quantum critical point.


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