$d$-wave Charge-$4e$ Superconductivity From Fluctuating Pair Density Waves

We present a theory for charge-$4e$ superconductivity as a leading low-temperature instability with a nontrivial $d$-wave symmetry. We show that in several microscopic models for the pair-density-wave (PDW) state, when the PDW wave vectors connect special parts of the Fermi surface, the predominant interaction is in the bosonic pairing channel mediated by exchanging low-energy fermions. This bosonic pairing interaction is repulsive in the $s$-wave channel but attractive in the $d$-wave one, leading to a $d$-wave charge-$4e$ superconductor. By analyzing the Ginzburg-Landau free energy including higher-order fluctuation effects of PDW, we find that the charge-$4e$ superconductivity emerges as a vestigial order of PDW, and sets in via a first-order transition. Both the gap amplitude and the transition temperature decay monotonically with increasing superfluid stiffness of the PDW order. Our work provides a microscopic mechanism of higher-charge condensates with unconventional ordering symmetry in strongly-correlated materials.