We investigate the effects of disorder on a system that in the clean limit is a pair density wave (PDW) superconductor. The charge order of the clean PDW is inevitably lost (via Imry-Ma), but the fate of the superconducting order is less clear. Here, we consider a strongly inhomogeneous limit in which the system consists of a random collection of PDW puddles embedded in a metallic background. When the puddles are dilute, they become phase coherent at low temperatures, resulting in a state that is macroscopically equivalent to a charge-$2e$ $s$-wave superconductor. This is a mechanism by which a PDW can have zero resistance, even in the presence of disorder, and is thermodynamically distinct from the “vestigial” charge-$4e$ superconductivity that has been proposed to arise in weakly disordered PDWs.

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• Charge density waves
• Impurities in superconductors
• Pair density wave
• Superconductivity
• Disordered systems