Anomalous superfluid density in pair-density-wave superconductors

Pair-density-wave (PDW) states are a long-sought-after phase of quantum materials, with the potential to unravel the mysteries of high-Tc cuprates and other strongly correlated superconductors. Yet, surprisingly, a key signature of stable superconductivity, namely the positivity of the superfluid density, ns(T), has not yet been demonstrated. Here, we address this central issue by calculating ns(T) for a generic model two-dimensional PDW superconductor. We uncover a surprisingly large region of intrinsic instability, associated with negative ns(T), revealing that a significant portion of the parameter space thought to be physical cannot support a pure PDW order. In the remaining stable regime, we predict two striking and observable fingerprints: a small longitudinal superfluid response and an unusual temperature dependence for ns(T). These generally model-independent, as well as experimentally relevant findings suggest that the fragility of the superfluid density poses a significant problem for the formation of stable, finite temperature PDW superconductivity.