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Monolayers of a prototypical cuprate high transition-temperature ($T_C$)superconductor $Bi_2Sr_2CaCu_2O_{8+\delta}$ (Bi2212) was recently found to show$T_C$ and other electronic properties similar to those of the bulk. Therobustness of superconductivity in an ideal two-dimensional (2D) system was anintriguing fact that defied the Mermin-Wagner theorem. Here, we took advantageof the high sensitivity of scanning SQUID susceptometry to image the phasestiffness throughout the phase transition of Bi2212 in the 2D limit. We foundsusceptibility oscillated with flux between diamagnetism and paramagnetism in aFraunhofer-like pattern up till $T_C$. The temperature and samplesize-dependence of the modulation period agreed well with our Coulomb gasanalogy of a finite 2D system based on Berezinskii-Kosterlitz-Thouless (BKT)transition. In the multilayers, the susceptibility oscillation differed in asmall temperature regime below $T_C$ in consistent with a dimensional-crossoverled by interlayer coupling. Serving as strong evidence of BKT transition in thebulk, there appeared a sharp superfluid density jump at zero-field andparamagnetism at small fields just below $T_C$. These results unified the phasetransitions from the monolayer Bi2212 to the bulk as BKT transition with finiteinterlayer coupling. This elucidating picture favored the pre-formed pairsscenario for the underdoped cuprates regardless of lattice dimensionality.

Oscillating paramagnetic Meissner effect and Berezinskii-Kosterlitz-Thouless transition in $Bi_2Sr_2CaCu_2O_{8+\delta}$ monolayer