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Feasibility of a Fulde-Ferrell-Larkin-Ovchinnikov superfluid Fermi atomic gas

Taira Kawamura and Yoji Ohashi
Phys. Rev. A 106, 033320 – Published 28 September 2022
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  • INTRODUCTION
  • FORMULATION
  • FEASIBILITY OF FFLO STATE IN…
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We theoretically explore a promising route to achieve the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in a spin-imbalanced ultracold Fermi gas. In the current stage of cold atom physics, search for this exotic Fermi superfluid is facing two serious difficulties: One is the desperate destruction of the FFLO long-range order by FFLO pairing fluctuations, which precludes entering the phase through a second-order transition, even in three dimension. The other is the fierce competition with the phase separation into the BCS (Bardeen-Cooper-Schrieffer) state and the spin-polarized normal state. By including strong FFLO pairing fluctuations within the framework of the strong-coupling theory developed by Nozières and Schmitt-Rink, we show that the anisotropy of Fermi surface introduced by an optical lattice makes the FFLO state stable against the paring fluctuations. This stabilized FFLO state is also found to be able to overcome the competition with the phase separation under a certain condition. Since the realization of unconventional Fermi superfluids is one of the most exciting challenges in cold atom physics, our results would contribute to the further development of this field.

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    • Received 28 June 2022
    • Accepted 13 September 2022

    DOI:https://doi.org/10.1103/PhysRevA.106.033320

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Cold atoms & matter wavesExotic phases of matterFFLOFermi gases
    1. Techniques
    Green's function methodsHubbard model
    Atomic, Molecular & Optical

    Authors & Affiliations

    Taira Kawamura* and Yoji Ohashi

    • Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

    • *tairakawa@keio.jp

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    Vol. 106, Iss. 3 — September 2022

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    Images

    • Figure 1
      Figure 1

      Fluctuation correction ΩFL to the thermodynamic potential Ω in the HNSR theory. The solid line and the dashed line are the 2×2 matrix mean-field single-particle thermal Green's function in Eq. (16) and the pairing interaction −U, respectively. Πij is the pair-correlation function in Eq. (27). The solid circle denotes a Pauli matrix τi in the Nambu representation.

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    • Figure 2
      Figure 2

      (a) Schematic superfluid phase diagram to explain how to determine phase boundaries. The solid (dashed) line represents an assumed second-order (first-order) superfluid phase transition temperature. When we approach the superfluid phase (BCS or FFLO state) from the normal state by decreasing the effective magnetic field h=[μ↑−μ↓]/2 along path (A) [path (B)] and pass through the second-order (first-order) phase transition line under the condition that the averaged chemical potential μ=[μ↑+μ↓]/2 is fixed, the thermodynamic potential ΩMF(Δ) as a function of Δ varies as shown in panel (b1) [panel (b2)]. In this case, as shown in panel (c1) [panel (c2)], the superfluid order parameter Δ¯ continuously (discontinuously) grows from zero in the superfluid phase below h=hc2nd (h=hc1st). In panels (b1) and (b2), the solid circle represents the value of Δ(>0) at which ΩMF(Δ) takes a local minimum. In the normal state, ΩMF(Δ) is always the smallest at Δ=0.

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    • Figure 3
      Figure 3

      Calculated second-order superfluid phase transition temperature Tc. In this figure we fix the total number N=N↑+N↓ of Fermi atoms. Tc is normalized by the value at P=0 (≡Tc0). (a) Fermi gas in the absence of optical lattice, when (kFas)−1=−1. (Here, as and kF are the s-wave scattering length and the Fermi momentum, respectively.) (b) Lattice Fermi gas, when U/(6t)=0.4, n=0.3, and t′=0. Panels (a1) and (b1) show the mean-field results (where pairing fluctuations are ignored). Panels (a2) and (b2) show the results including pairing fluctuations within the NSR and HNSR theories, respectively. The solid (dashed) line is the phase boundary between the BCS (FFLO) state and the normal state. Panels (c) and (d) show the intensity of the particle-particle scattering matrix Γ(q,iνn=0) in Eq. (34) at the positions (c) and (d) shown in panels (a2) and (b2), respectively. In panel (d), the lattice constant is taken to be unity (same for Figs. 4, 5, 6). The phase separation is ignored in this figure.

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    • Figure 4
      Figure 4

      Second-order superfluid phase transition temperature Tc in a lattice Fermi gas, as a function of the polarization P and the filling fraction n. (a1) HNSR theory. (b1) Mean-field BCS theory. The solid (dashed) line shows the BCS (FFLO) phase transition. The solid circle shows the Lifshitz point (LP), at which three kinds of phase boundaries between (1) the BCS and the normal states, (2) the FFLO and the normal states, as well as (3) the BCS and the FFLO states, meet one another. Panels (a2) and (b2) show the magnitude of the Q vector along the Tc line shown in panels (a1) and (b2), respectively. We set U/(6t)=0.4 and t′=0. Tc0 is the superfluid phase transition temperature when P=0.

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    • Figure 5
      Figure 5

      Assessment of the anisotropy of Fermi surface. We introduce an averaged Fermi sphere that has the same volume VFS as the anisotropic Fermi surface in a optical lattice [panel (a1)]. We then sum up the magnitude δVFS of the difference between these two Fermi surfaces at each momentum direction [panel (a2)]. (b1) Calculated δVFS/VFS, as a function of the filling fraction n. We set t′=0. (b2) δVFS/VFS as a function of the NNN hopping parameter t′, when n=0.4. For clarity, n dependence and t′ dependence of the Fermi surface shape at kz=0 are explicitly shown in panels (c1) and (c2), respectively.

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    • Figure 6
      Figure 6

      Same plots as Fig. 4 for various values of the NNN hopping amplitude t′, when n=0.4.

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    • Figure 7
      Figure 7

      (a) HNSR results on the second-order superfluid phase transition temperature Tc (solid line: BCS transition, dashed line: FFLO transition), as well as the phase-separation temperature TPS (dotted line). In each upper panel, the filling fraction n is fixed (n=0.3, 0.305, and 0.31). Below TPS, the system separates into the BCS and the spin-polarized normal states. (b) Thermodynamic potential ΩMF(Δ)−ΩMF(0), measured from the value at Δ=0. Each panel shows the Δ dependence along the path (b1)–(b3) shown in the upper panels. In panels (b1) and (b2), Δ¯(>0) is the position at which ΩMF(Δ) takes a local minimum (solid circle), and Pc is the polarization at the phase-separation line. Pc in panel (b3) denotes the polarization at the FFLO superfluid phase transition. We set U/(6t)=0.4 and t′=0.

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    • Figure 8
      Figure 8

      Same plots as Figs. 7(a1)–7(a3) for different values of NNN hopping t′/t, when n=0.4.

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    • Figure 9
      Figure 9

      Same plots as Fig. 7 in the T−h phase diagram. Since the filling fraction n is fixed in each panel, the phase separation occurs at TPS as in the T−P phase diagram in Fig. 7. Regarding this, we note that when h is decreased from the phase boundary at TPS under the condition that the averaged chemical potential μ is fixed, the BCS first-order phase transition occurs, without the phase separation.

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