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-$T_c$ cuprates and other strongly correlated superconductors. Yet, surprisingly, a key signature of stable superconductivity, namely the positivity of the superfluid density, $n_s(T)$, has not yet been demonstrated. Here, we address this central issue by calculating $n_s(T)$ for a generic model two-dimensional PDW superconductor. We uncover a surprisingly large region of intrinsic instability, associated with negative $n_s(T)$, 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 $n_s(T)$. 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.