Universal Long-distance Tail of Particle-number Density near a Wall in Superfluids

Abstract

We investigate the long-distance recovery of the particle-number density near a static, impermeable wall in a $d$-dimensional Galilean-invariant superfluid. Using superfluid effective field theory (EFT) on a half-space, we show that quantum phonon fluctuations produce a $z^{-(d+1)}$ tail in the smooth density profile at zero temperature, where $z$ is the distance from the wall. The tail coefficient is fixed by the bulk equation of state and is independent of microscopic wall details. At low temperature $T$, the thermal length $c_s/T$, with $c_s$ the bulk sound speed, separates the zero-temperature behavior at intermediate distances from a thermal $Tz^{-d}$ tail at larger distances for $d>1$. For $d=1$, the long-distance phonon contribution instead decays as $\exp(-4πTz/c_s)$. The EFT applies to both Bose and Fermi superfluids, including strongly interacting Fermi gases at unitarity. We estimate the normalized tail amplitude in three dimensions across the BCS--BEC crossover using experimental sound-speed data.