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Progress of Theoretical Physics

Superfluid Vortex, Trapping Neutral Impurities

Tetsuo Ohmi, Tunemaru Usui

Published Jun 1, 1969• 41 citations
Publisher

Abstract

Structure and energy of a superfluid vortex line and a neutral impurity (He3) trapped by it are investigated in the Hartree-Fock approximation. In dilute gas the shape of the impurity is prolate along the vortex line, while in dense gas it becomes more spherical. In liquid helium the ratio of the axes is estimated at √0.842 and the binding energy at 3.5°K. A rough estimate of the saturation density is given. It is pointed out that at this density the tension has decreased to 60% of that in pure He4.

DOI
10.1143/ptp.41.1401

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Structure and energy of a superfluid vortex line and a neutral impurity (He3) trapped by it are investigated in the Hartree-Fock approximation. In dilute gas the shape of the impurity is prolate along the vortex line, while in dense gas it becomes more spherical. In liquid helium the ratio of the axes is estimated at √0.842 and the binding energy at 3.5°K. A rough estimate of the saturation density is given. It is pointed out that at this density the tension has decreased to 60% of that in pure He4.
We calculate the binding energies of a 3He{}^{3}\mathrm{He}3He impurity and a fictitious 4He{}^{4}\mathrm{He}4He impurity to a superfluid 4He{}^{4}\mathrm{He}4He vortex. Within the accuracy of the calculation, there is no difference between the mass-3 and mass-4 binding energies, suggesting that the binding is due to the displacement of atoms from the core. Comparing the results between Onsager-Feynman ( ${\ensuremath{\Psi}}_{\mathrm{OF}}$) and ``shadow'' ( ${\ensuremath{\Psi}}_{D}$) variational functions, the energy is dependent on the density for ${\ensuremath{\Psi}}_{\mathrm{OF}}$, but independent for ${\ensuremath{\Psi}}_{D}$. The binding of 2.3 K for ${\ensuremath{\Psi}}_{D}$ at zero pressure is smaller than the experimental estimate. We also estimate the binding energies of charged ions.
Using many-body wave functions we have calculated the binding energies of a 3He impurity and a fictitious 4He impurity to a superfluid vortex. We use both a shadow wave function with distributed vorticity, ΨD, and the Onsagar–Feynman function with singular vorticity, ΨO−F. For both wave functions there was no difference in the binding energies of the mass 3 and mass 4 impurities. The binding energy was independent of the density for ΨD, but strongly density dependent for ΨO−F. The value of 2.3 K for ΨD, is somewhat smaller than the experimental estimate. Our results strongly suggest that the binding is due to the displacement of high-velocity 4He atoms from the vortex core by the impurity.
The magnitude, sign, and momentum dependence of the effective interaction between He3{\mathrm{He}}^{3}He3 atoms in superfluid He4{\mathrm{He}}^{4}He4 are derived from the experimental data. Calculations of the Fermi-liquid parameters for the normal state of He3{\mathrm{He}}^{3}He3 in solution yield results in agreement with experiment; the temperature for a superfluid transition associated with the He3{\mathrm{He}}^{3}He3 is estimated to be \ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{}K.
We compute the structure of a quantized vortex line in a harmonically trapped dilute atomic Bose-Einstein condensate using the Popov version of the Hartree-Fock-Bogoliubov mean-field theory. The vortex is shown to be (meta)stable in a nonrotating trap even in the zero-temperature limit, thus confirming that weak particle interactions induce for the condensed gas a fundamental property characterizing "classical" superfluids. We present the structure of the vortex at ultralow temperatures and discuss the crucial effect of the thermal gas component to its energetic stability.
The structure of an isolated vortex in a dilute two-component neutral superfluid Fermi gas is studied within the context of self-consistent Bogoliubov--de Gennes theory. Various thermodynamic properties are calculated, and the shift in the critical temperature due to the presence of the vortex is analyzed. The gapless excitations inside the vortex core are studied, and a scheme to detect these states and thus the presence of the vortex is examined. The numerical results are compared with various analytical expressions when appropriate.