Galilean invariance in confined quantum systems: implications for 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(⊥) leads to spectral gaps n(2)h(2)s(⊥)ρ/(2 mL) at wave numbers 2nπs(⊥)ρ, 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(⊥)ρ is kept fixed, translation invariance is broken, and the center of mass has a periodic distribution, while superfluidity persists at low flow velocities. This conclusion holds for the Lieb-Liniger model.