Parabolic Scaling in Overdoped Cuprate: a Statistical Field Theory Approach

Recently, Bozovic et al. reported that [Nature 536, 309-311 (2016)], in the overdoped side of the single-crystal $L{a}_{2-x}S{r}_{x}Cu{O}_4$ (LSCO) films, the transition temperature $T_c$ and zero-temperature superfluid phase stiffness ${\rho }_s(0)$ will obey a two-class scaling law: $T_c=\gamma \cdot \sqrt{\rho_s(0)}$for$T_c \leq T_Q$and$T_c \propto \rho_s(0)$for$T_c \geq T_M$,where$\gamma=(4.2 \pm 0.5) K^{1/2} $,$T_Q \approx 15 K$,and$T_M \approx 12 K$. They further pointed out that the parabolic scaling observed in the highly overdoped side indicates a quantum phase transition from a superconductor to a normal metal. In this paper, we propose a quantum partition function (QPF) for zero-temperature Cooper pairs, by which one can effectively distinguish between mean-field and quantum critical behaviors. We theoretically show that the two-class scaling law can be exactly derived by using the QPF, and the theoretical values of $\gamma$, $T_Q$, and $T_M$ are well in accordance with experimental measure values. Our analyses indicate that the linear scaling $T_c\propto {\rho }_s(0)$ is a mean-field behavior, while the parabolic scaling $T_c=\gamma \cdot \sqrt{{\rho }_s(0)}$ is a quantum critical behavior.