Enhancement of the London penetration depth in pnictides at the onset of SDW order under superconducting dome

Recent measurements of the doping dependence of the London penetration depth $\lambda(x)$ in clean samples of isovalent BaFe$_2$(As$_{1-x}$P$_x$)$_2$ at $T \ll T_c$ [Hashimoto et al., Science 336, 1554 (2012)] revealed a sharp peak in $\lambda(x)$ near optimal doping $x=0.3$. This observation points to the existence of the quantum critical point beneath the superconducting dome. We show that quantum magnetic fluctuations, associated with the emerging spin-density-wave order give rise to the observed feature. The effect comes from the dynamic renormalization of the effective mass $m^*$, which is related to $\lambda$ as $\lambda \propto \sqrt{m^*}$. We show that the effective mass has a maximum at the onset of the spin-density-wave order. We argue that the case of pnictides is conceptually different from a one-component Galilean invariant Fermi liquid, for which correlation effects do not cause the renormalization of the London penetration depth at $T=0$.