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The superfluid density of overdoped La$_{2-x}$Sr$_x$CuO$_4$ thin films ofhigh quality have been measured with Tc (defined by the onset of the Meissnereffect) from 5.1 to 41.6 K by Bosovic et al. Given this Tc the superfluiddensity shows no clear evidence of critical fluctuations and no indication ofvortex unbinding as T->Tc. Nevertheless, the superfluid density displays theexpected behavior for for a quantum phase transition (QPT) in the (3+1)D-xyuniversality class, rho(T=0)~Tc^2. However, this relation is also a hallmark ofdirty superconductors, treated in the mean-field approximation. Here we attemptto clear out the nature of the suppression of ro(T) as Tc->0. Noting that forany finite system the continuous transition will be rounded we perform a finitesize scaling analysis. It uncovers that the ro(T) data data are consistent witha finite length limited 3D-xy transition. In some films it is their thicknessand in others their inhomogeneity that determines the limiting length. Havingestablished the precondition for the occurrence of a QPT mapping on the (3+1)D-xy model, we explore the consistency with the hallmarks of this transition.In particular with the relations rho(T)/rho(0)=1-ycT/Tc, rho((0)^Tc^2,yc=alpha*Tc/ro(0), where alpha is the coefficient in rho(T)=rho(0)-alphaT asT->0. The emerging agreement with these characteristics points clearly to aquantum fluctuations induced suppression, revealing the crossover from thethermal to the quantum critical regime as Tc ->0. In the classical-quantummapping it corresponds to a 3D to (3 + 1)D crossover.

Quantum phase transition in overdoped La$_{2-x}$Sr$_x$CuO$_4$ evinced by the superfluid density