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

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 $c_s\mathrm{/}T$, with $c_s$ the bulk sound speed, separates the zero-temperature behavior at intermediate distances from a thermal $T{z}^{-d}$ tail at larger distances for $d>1$. For $d=1$, the long-distance phonon contribution instead decays as $\exp (-4\pi Tz\mathrm{/}{c}_s)$. 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.