Universal relationship between low-energy antiferromagnetic fluctuations and superconductivity in BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$

To identify the key parameter for optimal superconductivity in iron pnictides, we measured the $^{31}$P-NMR relaxation rate on BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ ($x = 0.22$ and 0.28) under pressure and compared the effects of chemical substitution and physical pressure. For $x = 0.22$, structural and antiferromagnetic (AFM) transition temperatures both show minimal changes with pressure up to 2.4~GPa, whereas the superconducting transition temperature $T_{rm c}$ increases to twice its former value. In contrast, for $x=0.28$ near the AFM quantum critical point (QCP), the structural phase transition is quickly suppressed by pressure and $T_{rm c}$ reaches a maximum. The analysis of the temperature-dependent nuclear relaxation rate indicates that these contrasting behaviors can be quantitatively explained by a single curve of the $T_{rm c}$ dome as a function of Weiss temperature $theta$, which measures the distance to the QCP. Moreover, the $T_{rm c}$-$theta$ curve under pressure precisely coincides with that with chemical substitution, which is indicative of the existence of a universal relationship between low-energy AFM fluctuations and superconductivity on BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$.