Thermodynamics of the Superfluid Dilute Bose Gas with Disorder

Abstract

We generalize the Beliaev-Popov diagrammatic technique for the problem of interacting dilute Bose gas with weak disorder. Averaging over disorder is implemented by the replica method. Low energy asymptotic form of the Green function confirms that the low energy excitations of the superfluid dirty Boson system are sound waves with velocity renormalized by the disorder and additional dissipation due to the impurity scattering. We find the thermodynamic potential and the superfluid density at any temperature below the superfluid transition temperature and derive the phase diagram in temperature vs. disorder plane.Superfluidity in random environments enjoys a long standing yet intense attention. The effect of disorder on the behavior of systems possessing long-range correlations is central to contemporary condensed matter physics, and superfluid Bose gas offers an exemplarily unique and accessible tool for both experimental and theoretical researches. One of the fascinating properties of such systems is their ability to maintain superfluidity (i.e. long range correlations) even in the strongly disordered environment. He 4 , for example, remains superfluid when absorbed in porous media [1]. The problem of influence of disorder on superfluidity (and on its close analogsuperconductivity) has been under extensive theoretical attack (see seminal works [2,3]) and remarkable progress in qualitative understanding of disordered Bose systems was achieved. Recent papers [4,5] discussed a continuum model of the dilute interacting Bose gas in a random potential. The advantages of this model are that (i) it is microscopically related to the original problem and (ii) it is very well understood in the clean limit. The proposed model describes, in particular, the quasiparticle dissipation and depletion of superfluidity at zero temperature and marked an important step towards quantitative description of disordered Bose systems.In this Letter, building on the model of Refs. [4,5], we develop a systematic diagrammatic perturbation theory for the dilute Bose gas with weak disorder at finite temperatures below the superfluid transition temperature T s . We obtain disorder corrections to the thermodynamic potential which completely determine thermodynamic properties of the superfluid system. We derive for the first time the disorder-induced shift of T s resulting from disorder scattering of quasiparticles with energy ǫ ∼ T . We find that the superfluid density decreases monotonically with the temperature. This completely agrees with the experimental data, while being in some contradiction with the theoretical result of Ref.[4] where a non-monotonic temperature dependence of superfluid density dependence was claimed. In the limit T → 0 our theory reproduces all the results of Refs. [4,5].The model. The starting point of our model is the Lagrangian density:where ϕ = ϕ(r, τ ) is the field representing Bose particles, r is the real space coordinate, τ is the Matsubara time and u(r) is the disorder potential. As usual we consider a soft intera...

Hartree–Fock mean-field theory for trapped dirty bosons

“…where, again, the normal ordering is explicitly emphasized and the second term is given by the tracelog of the integral kernel (27):…”

Dirty bosons in a three-dimensional harmonic trap

“…Also the disorderinduced shift of the critical temperature was analyzed in Refs. [28,29]. Furthermore, it was shown that dirty dipolar Bose gases yield characteristic directional dependences for thermodynamic quantities due to the emerging anisotropy of superfluidity at zero [31,32] and finite temperature [33][34][35].…”

Bogoliubov theory of dipolar Bose gas in a weak random potential

“…In order to study the properties of interacting bosons in such a random potential Huang and Meng proposed a Bogoliubov theory, which was applied to the case of superfluid helium in porous media [31], and extended later by others [32][33][34][35]. For a delta-correlated disorder it was found that both a condensate and a superfluid depletion occurs due to the localization of bosons in the respective minima of the external random potential which is present even at zero temperature.…”