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Finite-temperature superfluid depletion of disordered Bose gases

Cord A. Müller

Abstract

At zero temperature, homogeneous interacting Bose-condensed fluids are entirely superfluid, with remarkable transport properties. A non-superfluid, normal component is induced by finite temperatures and spatial inhomogeneity, the combined effects of which are rather intriguing, and difficult to describe quantitatively. By inhomogeneous Bogoliubov theory, applicable to weakly interacting condensed Bose gases in static external potentials with arbitrary spatial correlations, we calculate the normal fluid density via the transverse current-current correlation. We obtain finite-temperature disorder corrections to the normal fraction known since Laudau's seminal two-fluid theory, using diagrammatic perturbation theory for systems of any dimensionality, with closed analytical expressions to leading, quadratic order in disorder strength.

Finite-temperature superfluid depletion of disordered Bose gases

Abstract

At zero temperature, homogeneous interacting Bose-condensed fluids are entirely superfluid, with remarkable transport properties. A non-superfluid, normal component is induced by finite temperatures and spatial inhomogeneity, the combined effects of which are rather intriguing, and difficult to describe quantitatively. By inhomogeneous Bogoliubov theory, applicable to weakly interacting condensed Bose gases in static external potentials with arbitrary spatial correlations, we calculate the normal fluid density via the transverse current-current correlation. We obtain finite-temperature disorder corrections to the normal fraction known since Laudau's seminal two-fluid theory, using diagrammatic perturbation theory for systems of any dimensionality, with closed analytical expressions to leading, quadratic order in disorder strength.
Paper Structure (11 sections, 49 equations, 5 figures)

This paper contains 11 sections, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Starting lineup
  3. Single-bogolon contribution
  4. Pair-bogolon contribution
  5. Clean system: Landau's formula recovered
  6. Inhomogeneous system: corrections $O(V^2)$
  7. Summary and Outlook
  8. Continuity equation and longitudinal current response
  9. Number-conserving theory
  10. Bogoliubov theory
  11. Temperature-independent, pair-bogolon correction

Figures (5)

  • Figure 1: Single bogolon contribution to the normal fraction, eq. \ref{['fn1.eq']}, to first order in the disorder potential variance $v^2=V^2/\mu^2$, as function of reduced potential correlation length $\zeta = \sigma/\xi$ in dimensions $d=1,2,3$ (top to bottom). Dashed: Thomas-Fermi limit $f_\text{nTF}^{[1]}=v^2/d$.
  • Figure 2: Pair contributions to the current-current correlations: (a) normal contribution, Eq. \ref{['Phikernel']}. (b) anomalous contribution, Eq. \ref{['tildePhikernel']}.
  • Figure 3: (a) The current-current pair correlator of the clean system, Eq. \ref{['Phiclean']}, yields Landau's formula, Eq. \ref{['rhonLandau']}. Two types of impurity corrections are found: Type I, shown in (b): self-energy correction to the free normal propagator. Up to order $\mathcal{V}^2$, this does not affect the anomalous kernel, since $F_0$ on the other side vanishes. Type II: vertex corrections connect normal (c) and anomalous (d) propagators on opposite sides.
  • Figure 4: Finite-temperature normal fraction in the Thomas-Fermi regime of smooth potentials, Eq. \ref{['fnTF2']}, per potential variance $v^2=V^2/\mu^2$ and relative to the clean fraction, Eq. \ref{['fn0']}, as function of reduced temperature $k_\text{B} T/\mu$. Towards zero temperature, their ratio becomes constant, Eq. \ref{['fnTFlowT.eq']}, shown as dashed lines.
  • Figure 5: Temperature-independent pair-bogolon contribution to the normal fraction, Eq. \ref{['fnzero2.eq']}, per potential variance $v^2=V^2/\mu^2$ as function of reduced potential correlation length $\zeta = \sigma/\xi$ in dimensions $d=1,2,3$ (top to bottom).