Finite-temperature vortices in a rotating Fermi gas

S. N. Klimin, J. Tempere, N. Verhelst, and M. V. Milošević

Phys. Rev. A 94, 023620 – Published 16 August 2016

Abstract

Vortices and vortex arrays have been used as a hallmark of superfluidity in rotated, ultracold Fermi gases. These superfluids can be described in terms of an effective field theory for a macroscopic wave function representing the field of condensed pairs, analogous to the Ginzburg-Landau theory for superconductors. Here we establish how rotation modifies this effective field theory, by rederiving it starting from the action of Fermi gas in the rotating frame of reference. The rotation leads to the appearance of an effective vector potential, and the coupling strength of this vector potential to the macroscopic wave function depends on the interaction strength between the fermions, due to a renormalization of the pair effective mass in the effective field theory. The mass renormalization derived here is in agreement with results of functional renormalization-group theory. In the extreme Bose-Einstein condensate regime, the pair effective mass tends to twice the fermion mass, in agreement with the physical picture of a weakly interacting Bose gas of molecular pairs. Then we use our macroscopic-wave-function description to study vortices and the critical rotation frequencies to form them. Equilibrium vortex state diagrams are derived and they are in good agreement with available results of the Bogoliubov–de Gennes theory and with experimental data.

Received 27 November 2015

DOI:https://doi.org/10.1103/PhysRevA.94.023620

Physics Subject Headings (PhySH)

Fermi gases

Ultracold gases Vortices in superfluids

Atomic, Molecular & Optical

Authors & Affiliations

S. N. Klimin^*, J. Tempere^†, and N. Verhelst

TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium

M. V. Milošević

Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium

^*Also at Department of Theoretical Physics, State University of Moldova, Chisinau 2009, Moldova.
^†Also at Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA.

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Issue

Vol. 94, Iss. 2 — August 2016

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Images

Figure 1
(a) Inverse renormalization factor $1 / \tilde{e}$ as a function of the dimensionless inverse scattering length $1 / k_{F} a_{s}$ and the relative temperature $T / T_{c}$ for a three-dimensional Fermi gas in a cylindrically symmetric parabolic confinement potential, with the number of particles per unit length $N = 1000$ . (b) Inverse renormalization factor obtained within the functional renormalization-group theory [40] for $T = 0$ . The renormalization factor $\tilde{e}$ is associated with the effective pair mass, as shown in the Appendix.
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Figure 2
Radius of the superfluid state as a function of the rotation frequency for a rotating Fermi gas with $1 / a_{s} = 0$ and $N = 10^{3}$ confined to a cylindrically symmetric parabolic potential, at different temperatures. The inset shows the half-distance between vortex centers for a vortex pair at two temperatures.
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Figure 3
Area of existence for vortices for a Fermi gas trapped in a cylindrically symmetric parabolic potential at $T = 0.01 T_{F}$ , with different numbers of particles per unit length.
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Figure 4
The left-hand axis (indicated by the arrow for two lower curves) shows the lower critical rotation frequency $ω_{c, 1}$ (in units of the trapping potential parameter $ω_{0}$ ) for a trapped Fermi gas as a function of the number of particles per unit length for $1 / k_{F} a_{s} = 0$ , at two temperatures $k_{B} T = 0.01 E_{F}$ and $0.1 E_{F}$ . The right-hand axis (indicated by the arrow for two upper curves) shows the ratio $ω_{c} / ω_{B}$ , where $ω_{B}$ is given by formula (18).
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Figure 5
Equilibrium vortex state diagram for a trapped rotating Fermi gas in a cylindrically symmetric parabolic confinement potential, showing the critical rotation frequencies as a function of the inverse scattering length for $T = 0.1 T_{F}$ and the number of particles per unit length $N = 10^{3}$ . The critical rotation frequencies are plotted for a single vortex and for a vortex pair. Also shown is the upper bound for the rotation frequency, which restricts the area of existence for the superfluid state.
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Figure 6
Equilibrium vortex state diagrams for a trapped rotating Fermi gas in a cylindrically symmetric parabolic confinement potential, showing the critical rotation frequencies as a function of the temperature for two numbers of particles per unit length and three inverse scattering lengths (indicated in the figure). The notation is the same as in Fig. 5. Symbols in (d) show the critical rotation frequencies $ω_{c, 1}$ (circles) and $ω_{c, 2}$ (squares) from the Supplemental Material to Ref. [23].
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