Galilean invariance in confined quantum systems: Implications on spectral gaps, superfluid flow, and periodic order

Galilean invariance leaves its imprint on the energy spectrum and eigenstates of N quantum particles, bosons or fermions, confined in a bounded domain. It endows the spectrum with a recurrent structure which in capillaries or elongated traps of length L and cross-section area s_perp leads to spectral gaps n^2h2s_perpρ/(2mL) at wavenumbers 2nπs_perpρ, where ρ is the number density and m is the particle mass. In zero temperature superfluids, in toroidal geometries, it causes the quantization of the flow velocity with the quantum h/(mL) or that of the circulation along the toroid with the known quantum h/m. Adding a "friction" potential which breaks Galilean invariance, the Hamiltonian can have a superfluid ground state at low flow velocities but not above a critical velocity which may be different from the velocity of sound. In the limit of infinite N and L, if N/L=s_perpρ is kept fixed, translation invariance is broken, the center of mass has a periodic distribution, while superfluidity persists at low flow velocities. This conclusion holds for the Lieb-Liniger model.