The Gross-Pitaevskii approximation is a long-wavelength theory widely used to describe a variety of properties of dilute Bose condensates, in particular trapped alkali gases. We point out that for short-ranged repulsive interactions this theory fails in dimensions d</=2, and we propose the appropriate low-dimensional modifications, which have a universal form. For d = 1 we analyze density profiles in confining potentials, superfluid properties, solitons, and self-similar solutions.

arXiv:cond-mat/0002282Physical Review LettersAug 7, 2000319 citations

Correlation functions for one-dimensional bosons at low temperature

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Annals of Physics

Theory of a low-density bose liquid

D J Amit, Eugene P. Gross

Published Apr 1, 1967• 2 citations

Publisher

DOI: 10.1016/0003-4916(67)90072-3

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n(x)

, the pair distribution function

g(x)

, and the static structure factor

S(q)

are evaluated. The aim is to test Luttinger liquid (LL) predictions for 1D fluids over a wide range of fluid density and LL parameter

K

. The 1D Bose fluid examined is a single chain of

^{4}\mathrm{He}

atoms confined to a line in the center of a narrow nanopore. The atoms cannot exchange positions in the nanopore, the criterion for 1D. The fluid density is varied from the spinodal density where the 1D liquid is unstable to droplet formation to the density of bulk liquid

^{4}\mathrm{He}

. In this range,

K

varies from

K>2

at low density, where a robust superfluid is predicted, to

K<0.5

, where fragile 1D superflow and solidlike peaks in

S(q)

are predicted. For uniform pore walls, the ${\ensuremath{\rho}}_{S}/{\ensuremath{\rho}}_{0}$ scales as predicted by LL theory. The

n(x)

and

g(x)

show long range oscillations and decay with

x

as predicted by LL theory. The amplitude of the oscillations is large at high density (small

K)

and small at low density (large

K)

. The

K

values obtained from different properties agree well verifying the internal structure of LL theory. In the presence of disorder, the ${\ensuremath{\rho}}_{S}/{\ensuremath{\rho}}_{0}$ does not scale as predicted by LL theory. A single

{v}_{J}

parameter in the LL theory that recovers LL scaling was not found. The one body density matrix (OBDM) in disorder is well predicted by LL theory. The ``dynamical'' superfluid fraction, ${\ensuremath{\rho}}_{S}^{D}/{\ensuremath{\rho}}_{0}$, is determined. The physics of the deviation from LL theory in disorder and the ``dynamical'' ${\ensuremath{\rho}}_{S}^{D}/{\ensuremath{\rho}}_{0}$ are discussed.