Galilean Invariance in Confined Quantum Systems: Implications for Spectral Gaps, Superfluid Flow, and Periodic Order

Abstract

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/(2mL) 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, the center of mass has a periodic distribution, while superfluidity persists at low flow velocities. This conclusion holds for the Lieb-Liniger model.

The total momentum of quantum fluids

“…[In a supersolid one would have the convex combination of an atomic and a continuous probability distribution.] This corresponds to what we found in one dimension at zero temperature [5]. Because we proved earlier that in two dimensions at all temperatures and densities the total momentum diverges as √ N with a nonvanishing probability, cf.…”

“…In Theorem III.1 first we show that in any eigenstate of the total momentum operator the center of mass can be separated and it propagates as a free particle carrying the total momentum. This result, applied already in our earlier paper [5], is intuitively obvious but its derivation needs some care. We then use this separability to define ρ c.m.…”

“…We think this to be typical for crystals; see also the result of the paper [5] for the ground state in 1D and the forthcoming discussion in Section VII.…”

“…The restriction to T > 0 is crucial. The ground state can be periodically ordered [5], and there certainly exist low-lying excited states which correspond to its periodic modulations, i.e., to density waves. The absence of density waves at T > 0 is the synonym of the absence of structural phase transitions.…”

“…We then use Galilean invariance to sum separately over eigenstates belonging to the irreducible wave vectors, introduced in Ref. [5], and over their Galilean boosts. This leads to some interesting results.…”