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Physical Review Letters

Core Structure of a Vortex in Superfluid He4

Gerardo Ortíz, David M. Ceperley

Published Dec 18, 1995• 50 citations
Publisher

Abstract

The microscopic structure of a quantized vortex core in 2D $^{4}\mathrm{He}$ is investigated using the fixed-phase approach. We present a general method to suggest new forms for the phase of a many-particle wave function. Starting from the Feynman phase, our scheme generates backflow corrections with qualitatively different properties than the starting Feynman vortex. We evaluate particle density, circulating current, core radius, and energy for an isolated vortex at zero temperature using Green's function Monte Carlo.

DOI
10.1103/physrevlett.75.4642

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The microscopic structure of a quantized vortex core in 2D 4He^{4}\mathrm{He}4He is investigated using the fixed-phase approach. We present a general method to suggest new forms for the phase of a many-particle wave function. Starting from the Feynman phase, our scheme generates backflow corrections with qualitatively different properties than the starting Feynman vortex. We evaluate particle density, circulating current, core radius, and energy for an isolated vortex at zero temperature using Green's function Monte Carlo.
We present a diffusion Monte Carlo study of a vortex line excitation attached to the center of a 4He^{4}\mathrm{He}4He droplet at zero temperature. The vortex energy is estimated for droplets of increasing number of atoms, from N=70N=70N=70 up to 300, showing a monotonous increase with NNN. The evolution of the core radius and its associated energy, the core energy, is also studied as a function of NNN. The core radius is $\ensuremath{\sim}1\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ in the center and increases when approaching the droplet surface; the core energy per unit volume stabilizes at a value $2.8\phantom{\rule{0.3em}{0ex}}\mathrm{K}{\ensuremath{\sigma}}^{\ensuremath{-}3}$ $(\ensuremath{\sigma}=2.556\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$ for $N\ensuremath{\ge}200$.
In two dimensions a microscopic theory providing a basis for the naive analogy between a quantized vortex in a superfluid and an electron in a uniform magnetic field is presented. Following the variational approach developed by Peierls, Yoccoz, and Thouless, the cyclotron motion of a vortex is described by the many-body wave function, which is a linear combination of Feynman wave functions centered at different positions. An integral equation for the weighting functions of the superposition is derived by minimizing the energy functional. The matrix elements of the kernel are the overlaps between any two displaced Feynman wave functions. A numerical study is conducted for a bosonic superfluid based on a Hartree ground state. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with energy, and scales with the size of the vortex. The fluid density is finite at the vortex axis and the vorticity is distributed in the core region. The effective mass of a quantized vortex defined by the inverse of the energy-level spacing is shown to be logarithmically divergent with the size of the vortex.
Model many-body wave functions are constructed for a quantized vortex line and ring. The corresponding variational energies are calculated using an approximate integral equation derived by Percus and Yevick for classical fluids. The energy of the line is close to that found in the Hartree theory. However, the density distribution in the core region is markedly different, and the core size is somewhat smaller, being of the order of 1 \AA{}. The velocity of translation of the vortex ring is calculated by a new method, and excellent agreement is obtained with the experimental results of Rayfield and Reif. The condensed-state wave function is computed numerically, and it is found that in the core of the vortex there are about 20% fewer particles in the condensed state than in the zero-momentum state when the system is in equilibrium.
In order to understand what kind of information can be extracted from the vortex-core structure in superfluid fermion-atom gases we make a systematic study for the core structure of a singly quantized vortex in the range of broad Feshbach resonance, solving numerically the generalized Bogoliubov--de Gennes equation derived from the fermion-boson model. Numerical calculations reveal that the vortex core can be well characterized by three length scales from the spatial variation of the fermionic and bosonic gap functions. We also predict that the profile of matter depletion and the quasiparticle spectrum show distinctive differences between the BCS and Bose-Einstein Condensate regimes.
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.