Hard-sphere Bose gas in random external potentials

Abstract

We consider a dilute hard-sphere Bose gas in random external potentials at low temperatures, in Z) = 3, using the technique of pseudopotentials and the Bogliubov transformation. At absolute zero, the random potentials can deplete the Bose condensate, though not completely. On the other hand, they generate an amount of normal fluid equal to j the condensate depletion. This is a localization effect that can destroy superfluidity at absolute zero. General features of the superfluid density in the neighborhood of this transition point agree qualitatively with experimental results on helium in porous media.PACS numbers: 67.40. Bz, 05.30.Jp, 64.60.Cn We report on some results concerning the low-temperature properties of a dilute hard-sphere Bose gas in random external potentials in three dimensions. Such a model is a crude simulation of superfluid helium in porous media [1]. The spongelike media are here idealized as random distributions of hard-sphere potentials. To make the problem tractable, we further assume that the randomness is sufficiently dilute, and the temperature sufficiently low, that the hard spheres can be approximated by delta-function pseudopotentials [2]. We also assume that the potentials are distributed with uncorrelated randomness. Thus, the very large pores that are apparently present in the experimental media are not taken into account here. The purpose of this study is not to construct a quantitative model for the experiments, but to illuminate some qualitative features. We are able to show, for example, that at absolute zero superfluidity can where y/Cx) is the field operator for nonrelativistic boson of mass ra, N=fd 3 xy/^y/ is the number operator, U(x) is the external potential, and v^^Ana/m, where a is the hard-sphere diameter. The ground-state energy is rendered finite by subtracting an appropriate divergent constant [3].The external potential U(x) may be pictured as a sum of randomly located scattering centers of random strengths, either attractive or repulsive. We assume (U(x)U(y)) av oz8 3 (x-y), and characterize the potentials by a single parameter Ro,where V is the total volume of the system, £/* is the Fourier transform of U(x), and the subscript av denotes a quenched average over potentials. It has dimension (energy) 2 (length) 3 , and is (average density)x(meansquare strength) of the individual scatterers, the strength be destroyed by the randomness, through an effect suggestive of boson localization. From a theoretical point of view, the interparticle interactions are necessary to prevent a total condensation into a single localized orbital in the external potential. Thus, unlike the much-studied case of fermions, one does not have the luxury of treating the potentials as perturbations on a free-particle Hamiltonian. Here we do the next best thing, namely, start with the simplest soluble problem involving interparticle interactions-a dilute hard-sphere gas at low temperatures [2,3]. This approach differs from previous efforts on this subject [4,5], in that ours is a microscopic...

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