Normal charge densities in cold quantum critical superfluids

Recent experiments on overdoped LSCO by Bozovic et al. [Nature 536, 282 (2016)] have revealed an anomalously low superfluid density. Further measurements by Mahmood et al. [PRL 122, 027003 (2019)] on the low-temperature optical conductivity tie this to a large residual normal density, at odds with expectations from BCS theory and general arguments due to Legget [J. Stat. Phys 93, 927 (1998)]. We revisit these arguments in the context of superfluid hydrodynamics and demonstrate that consistently coupling the system to external sources allows for a large low-temperature normal density. As a proof-of-principle, we use gauge-gravity duality to construct an example of a strongly-coupled superfluid which has non-vanishing zero-temperature normal and superfluid densities. We find that the coexistence of the two densities at zero-temperature is a consequence of the underlying quantum critical ground state and the spectrum of irrelevant deformations around this state. Our results are in qualitative agreement with the optical conductivity as well as other previous measurements on overdoped LSCO, including the specific heat.