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Phase transition beneath the superconducting dome in BaFe$_2$(As$_{1-x}$P$_x$)$_2$

We present a theory for the large suppression of the superfluid-density, $ρ_s$, in BaFe$_2$(As$_{1-x}$P$_x$)$_2$ in the vicinity of a putative spin-density wave quantum critical point at a P-doping, $x=x_{c}$. We argue that the transition becomes weakly first-order in the vicinity of $x_{c}$, and disorder induces puddles of superconducting and antiferromagnetic regions at short length-scales; thus the system becomes an electronic micro-emulsion. We propose that frustrated Josephson couplings between the superconducting grains suppress $ρ_s$. In addition, the presence of `normal' quasiparticles at the interface of the frustrated Josephson junctions will give rise to a highly non-trivial feature in the low frequency response in a narrow vicinity around $x_c$. We propose a number of experiments to test our theory.