Evidence for Kosterlitz-Thouless and three-dimensional $XY$ critical behavior in ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$

We present reversible magnetization data of a high-quality ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ single crystal and explore the occurrence of three-dimensional (3D)-$xy$ critical behavior close to the bulk-transition temperature $T_c$ and of Kosterlitz-Thouless (KT) behavior. Below and above the presumed Kosterlitz-Thouless transition temperature $T_{\text{KT}}$ we observe the characteristic two-dimensional (2D)-$xy$ behavior: a downward shift in the crossing point phenomenon toward $T_{\text{KT}}$ as the field is decreased and sufficiently below $T_{\text{KT}}$ the characteristic 2D-$xy$ relationship between the magnetization and the in-plane magnetic penetration depth ${\lambda }_{ab}$. In contrast, the measured temperature dependence of the superfluid density does not exhibit the characteristic KT behavior around the presumed $T_{\text{KT}}$. The absence of this feature is traced back to the 2D- to 3D-$xy$ crossover setting in around and above $T_{\text{KT}}$. Invoking the Maxwell relation, the anomalous field dependence of the specific-heat peak is also traced back to the intermediate 2D-$xy$ behavior. However, close to $T_c$ we observe consistency with 3D-$xy$ critical behavior in agreement with measurements of ${\lambda }_{ab}$.