BCS quantum critical phenomena

Theoretically, we recently showed that the scaling relation between the transition temperature T_c and the superfluid density at zero temperature n_s (0) might exhibit a parabolic pattern [Scientific Reports 6 (2016) 23863]. It is significantly different from the linear scaling described by Homes' law, which is well known as a mean-field result. More recently, Bozovic et al. have observed such a parabolic scaling in the overdoped copper oxides with a sufficiently low transition temperature T_c [Nature 536 (2016) 309-311]. They further point out that this experimental finding is incompatible with the standard Bardeen-Cooper-Schrieffer (BCS) description. Here we report that if T_c is sufficiently low, applying the renormalization group approach into the BCS action at zero temperature will naturally lead to the parabolic scaling. Our result indicates that when T_c sufficiently approaches zero, quantum fluctuations will be overwhelmingly amplified so that the mean-field approximation may break down at zero temperature.