Anomalous dimensional crossover in critical microwave-conductivity fluctuations of superconducting La$_{2-x}$Sr$_{x}$CuO$_{4}$ thin films

We demonstrate that there are two dimensional crossover lines separating the phase diagram of La$_{2-x}$Sr$_{x}$CuO$_{4}$ (LSCO) into three regions with different universality classes, by using a dynamic scaling analysis of the microwave complex conductivity. For underdoped LSCO from x=0.07 to x=0.14, we show clear evidence for the 2D-\textit{XY} universality class (the BKT transition in the nearly decoupled CuO$_{2}$ planes), while the 3D-\textit{XY} universality class is observed for nearly optimally doped region (x=0.15, 0.16). Surprisingly, for overdoped LSCO with x$>$0.17, we found that the critical behavior strongly suggested the 2D universality class, in contrast to the reduction of anisotropic properties with hole-doping. We discuss the implication of these results in terms of the effect of the quantum critical fluctuations.