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Journal of Physics Condensed Matter

Bosons in a random potential: condensation and screening in a dense limit

Derek K. K. Lee, J M F Gunn

Published Sep 24, 1990• 47 citations
Publisher

Abstract

The authors demonstrate that an extended Bose condensate can be stable in a random potential for a suitable weak-repulsive limit of a dense Bose gas, even though the non-interacting case is pathological. The condensate exists primarily because the interactions allow screening of the random potential. This may happen even when the chemical potential is in the Lifshitz tails of the single-particle case. Indeed, the authors argue that there are no Lifshitz tail states in their dense but weakly-interacting system. Using a number-phase representation, they calculate the increase in the depletion of the condensate with increasing randomness (at fixed density) which indicates the eventual destruction of the condensed phase-perhaps to a localized phase. The physical picture discussed should be relevant to the understanding of helium thin films.

DOI
10.1088/0953-8984/2/38/004

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AIP conference proceedings•Jan 1, 2009•0 citations
The authors demonstrate that an extended Bose condensate can be stable in a random potential for a suitable weak-repulsive limit of a dense Bose gas, even though the non-interacting case is pathological. The condensate exists primarily because the interactions allow screening of the random potential. This may happen even when the chemical potential is in the Lifshitz tails of the single-particle case. Indeed, the authors argue that there are no Lifshitz tail states in their dense but weakly-interacting system. Using a number-phase representation, they calculate the increase in the depletion of the condensate with increasing randomness (at fixed density) which indicates the eventual destruction of the condensed phase-perhaps to a localized phase. The physical picture discussed should be relevant to the understanding of helium thin films.
The properties of systems with Bose-Einstein condensate in external time-independent random potentials are investigated in the frame of a self-consistent stochastic mean-field approximation. General considerations are presented, which are valid for finite temperatures, arbitrary strengths of the interaction potential, and for arbitrarily strong disorder potentials. The special case of a spatially uncorrelated random field is then treated in more detail. It is shown that the system consists of three components, condensed particles, uncondensed particles, and a glassy density fraction, but that the pure Bose glass phase with only a glassy density does not appear. The theory predicts a first-order phase transition for increasing disorder parameter, where the condensate fraction and the superfluid fraction simultaneously jump to zero. The influence of disorder on the ground-state energy, the stability conditions, the compressibility, the structure factor, and the sound velocity are analyzed. The uniform ideal condensed gas is shown to be always stochastically unstable, in the sense that an infinitesimally weak disorder destroys the Bose-Einstein condensate, returning the system to the normal state; but the uniform Bose-condensed system with finite repulsive interactions becomes stochastically stable and exists in a finite interval of the disorder parameter.
We determine the extent to which Lifshitz tail states can be defined in a condensed Bose system. This provides a justification for the inert-layer picture in the study of helium on Vycor. We argue that, in three dimensions, hard-core bosons can be trapped by a sufficiently deep potential well despite a strong tendency for all particles to participate in the Bose-Einstein condensate. This is demonstrated by a variant of the conventional spin-boson problem. We find that phase fluctuations in the condensate tend to depress particle exchange between the trap and the rest of the system and hence has an enhancing effect in terms of Lifshitz-tail localization. However at low densities (near onset of condensation) we argue that the condensate will be relatively enhanced compared to the inert layer model, in agreement with experiment.
A perfect Bose gas can condensate in one dimension in the presence of a random potential due to the presence of Lifshitz tails in the one-particle density of states. Here, we show that scale-free correlations in the random potential suppress the disorder induced Bose-Einstein condensation (BEC). Within a tight-binding approach, we consider free Bosons moving in a scale-free correlated random potential with spectral density decaying as $1/{k}^{\ensuremath{\alpha}}$. The critical temperature for BEC is shown to vanish in chains with a binary nonstationary potential $(\ensuremath{\alpha}>1)$. On the other hand, a weaker suppression of BEC takes place in nonbinarized scale-free potentials. After a slightly increase in the stationary regime, the BEC transition temperature continuously decays as the spectral exponent $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$.
A model of the mean-field interacting boson gas trapped by a weak harmonic potential is considered by the boson random point field methods. We prove that in the weak harmonic trap limit there are two phases distinguished by the boson condensation and by a different behavior of the local particle density. For chemical potentials less than a certain critical value, the resulting random point field (RPF) coincides with the usual boson RPF, which corresponds to a noninteracting (ideal) boson gas. For the chemical potentials greater than the critical value, the boson RPF describes a divergent (local) density, which is due to localization of the macroscopic number of condensed particles. Notice that it is this kind of transition that observed in experiments producing the Bose–Einstein condensation in traps.
The localization-disorder paradigm is analyzed for a specific system of weakly repulsive Bose gas at zero temperature placed into a quenched random potential. We show that at low average density or weak enough interaction the particles fill deep potential wells of the random potential whose radius and depth depend on the characteristics of the random potential and the interacting gas. The localized state is the random singlet with no long-range phase correlation. At a critical density the quantum phase transition to the coherent superfluid state proceeds. We calculate the critical density in terms of the geometrical characteristics of the noise and the gas. In a finite system the ground state becomes nonergodic at very low density. For atoms in traps four different regimes are found; only one of it is superfluid. The theory is extended to lower (one and two) dimensions. Its quantitative predictions can be checked in experiments with ultracold atomic gases and other Bose systems.
The localization‐disorder paradigm is analyzed for a specific system of weakly repulsive Bose gas at zero temperature placed into a quenched random potential. We show that, at low average density or weak enough interaction the particles fill deep potential wells of the random potential whose radius and depth depend on the characteristics of the random potential and the interacting gas. The localized state is the random singlet with no long‐range phase correlation. At a critical density the quantum phase transition to the coherent superfluid state proceeds. We calculate the critical density in terms of the geometrical characteristics of the noise and the gas. In a finite system the ground state becomes non‐ergodic at very low density. For atoms in traps four different regimes are found, only one of it is superfluid. The theory is extended to lower (1 and 2) dimensions. Its quantitative predictions can be checked in experiments with ultracold atomic gases and other Bose‐systems.This article is an extended version of a presentation delivered by one of the authors (V.L.P.) at a Conference on Modern Trends in Theoretical Physics dedicated to the Lev Landau Centenary (Chernogolovka, Russia, June 2008).