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Stable nonequilibrium Fulde-Ferrell-Larkin-Ovchinnikov state in a spin-imbalanced driven-dissipative Fermi gas loaded on a three-dimensional cubic optical lattice

Taira Kawamura, Daichi Kagamihara, and Yoji Ohashi
Phys. Rev. A 108, 013321 – Published 31 July 2023
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  • INTRODUCTION
  • FORMULATION
  • STABILIZATION OF THE NFFLO STATE IN…
  • EFFECTS OF SPIN IMBALANCE: RELATION TO…
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We theoretically investigate a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type superfluid phase transition in a driven-dissipative two-component Fermi gas. The system is assumed to be in the nonequilibrium steady state, which is tuned by adjusting the chemical potential difference between two reservoirs that are coupled with the system. Including pairing fluctuations by extending the strong-coupling theory developed in the thermal-equilibrium state by Nozières and Schmitt-Rink to this nonequilibrium case, we show that a nonequilibrium FFLO (NFFLO) phase transition can be realized without spin imbalance, under the conditions that (1) the two reservoirs imprint a two-edge structure on the momentum distribution of Fermi atoms and (2) the system is loaded on a three-dimensional cubic optical lattice. While the two edges work like two Fermi surfaces with different sizes, the role of the optical lattice is to prevent the NFFLO long-range order from destruction by NFFLO pairing fluctuations. We also draw the nonequilibrium mean-field phase diagram in terms of the chemical potential difference between the two reservoirs, a fictitious magnetic field to tune the spin imbalance of the system, and the environmental temperature of the reservoirs to clarify the relation between the NFFLO state and the ordinary thermal-equilibrium FFLO state discussed in spin-imbalanced Fermi gases.

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    • Received 15 May 2023
    • Accepted 19 July 2023

    DOI:https://doi.org/10.1103/PhysRevA.108.013321

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    FFLOFermi gasesFermi surface
    1. Physical Systems
    Fermionic condensatesNon-Hermitian systems
    1. Techniques
    Nonequilibrium Green's function
    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

    Authors & Affiliations

    Taira Kawamura1,*, Daichi Kagamihara2, and Yoji Ohashi1

    • 1Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
    • 2Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502, Japan

    • *tairakawa@keio.jp

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    Vol. 108, Iss. 1 — July 2023

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    Images

    • Figure 1
      Figure 1

      (a) Cooper pair in the thermal-equilibrium FFLO state in a spin-imbalanced Fermi gas. The large (small) colored circle with the radius pF↑ (pF↓) represents the Fermi surface of the ↑-spin (↓-spin) component. The dotted line with two small circles at the ends denotes a FFLO Cooper pair. (b) NFFLO Cooper pairs. The color intensity schematically describes the particle occupancy in momentum space and the edges at pFL and pFR (where the occupancy sharply changes) work like two Fermi surfaces with different radius. In this case, besides the conventional BCS-type Cooper pairs with zero center-of-mass momentum (dashed lines), the FFLO-type Cooper pairs (dotted lines) also become possible. In the latter case, Cooper pairs are formed between an ↑-spin Fermi atom near the edge at pFL and a ↓-spin atom near the edge at pFR, as well as an ↑-spin atom near the edge at pFR and an ↓-spin atom near the edge at pFL. (c) Particle occupation in the momentum space in a spin-imbalanced driven dissipative Fermi gas. When we simply call each edge “Fermi surface,” this system looks as if there are four Fermi surfaces at pF↑L, pF↑R, pF↓L, and pF↓R.

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    • Figure 2
      Figure 2

      (a) Model driven-dissipative two-component (σ=↑,↓) ultracold Fermi gas with a tunable s-wave pairing interaction −U (<0). The central main system is coupled with two reservoirs (α=L,R) in the thermal equilibrium state, having different values of the Fermi chemical potentials, μL=μ+δμ and μR=μ−δμ. Both reservoirs are free Fermi gases at the common environment temperature Tenv. Λα denotes a tunneling matrix element between the main system and the α reservoir. When δμ≠0, the momentum distribution np,σ of Fermi atoms has two edges around pFL=2mμL and pFR=2mμR at low temperatures (where m is an atomic mass), as shown in panel (b). These edges correspond to the Fermi surface edges illustrated in Fig. 1.

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    • Figure 3
      Figure 3

      Energy band in the main system, as well as those in the left and right reservoirs. The energy is commonly measured from the bottom (ɛk=0) of the band in the main system. −ωα gives the bottom of the energy band in the α reservoir. In the α reservoir at Tenv=0, the σ-spin band is filled up to the Fermi chemical potential μα,σ, given in Eqs. (8) and (9). We note that, when the reservoirs are spin imbalanced (h≠0), the main system is also spin imbalanced.

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    • Figure 4
      Figure 4

      (a) Dyson equation for the 2×2 matrix single-particle nonequilibrium Green's function ĜNMF,σ (double solid line) in the main system. The self-energy Σ̂NMF,σ describes effects of the pairing interaction −U (wavy line) within the NMF level. The thick solid line denotes Ĝenv,σ, which obeys the other Dyson equation in panel (b). The self-energy Σ̂env,σ involves effects of system-reservoir couplings within the second-order Born approximation. In panel (b), Green's functions Ĝ0,σ and D̂0,σα, respectively, describe free lattice fermions in the main system and a free Fermi gas in the α reservoir.

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    • Figure 5
      Figure 5

      (a) Nonequilibrium 2×2 particle-particle scattering matrix Γ̂ in Keldysh space. The double solid line is ĜNMF,σ given in Eq. (19). (b) Truncated Dyson equation giving the NNSR single-particle Green's function ĜNNSR,σ (thick solid line) in the main system. The self-energy Σ̂NNSR,σ describes effects of pairing fluctuations.

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    • Figure 6
      Figure 6

      Calculated Tenvc (upper panels) and |QFF| (lower panels) in a driven-dissipative spin-balanced lattice Fermi gas, as functions of the chemical potential bias δμ and the damping rate γ. (a) NMF theory. (b) NNSR theory. The solid (dashed) line is the phase boundary between the normal state and the NBCS (NFFLO) state with |QFF|=0 (|QFF|>0). We take t′=0, n=n↑+n↓=0.3, and U/(6t)=0.8. (Note that 6t is the bandwidth in the main system, when t′=0.) The thermal equilibrium limit is at δμ=0 and γ→+0.

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    • Figure 7
      Figure 7

      Same plots as the upper panels in Fig. 6, in the case when the optical lattice is absent. We set (pFas)−1=−0.6, where as is the s-wave scattering length. pF and ɛF as ɛF=pF2/(2m) are, respectively, the Fermi momentum and the Fermi energy of a free Fermi gas with the particle number N=pF3/(3π2). In panel (b), Tenvc is seen to exhibit reentrant behavior, due to the complete destruction of the NFFLO long-range order by anomalously enhanced NFFLO pairing fluctuations [52, 53].

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    • Figure 8
      Figure 8

      Calculated Tenvc as a function of δμ. We set t′=0, n=n↑+n↓=0.3, γ/(6t)=0.005, and U/(6t)=0.8. In each NMF and NNSR result, the solid circle is the boundary between the NBCS and NFFLO phase transitions. The inset shows the results when Tenvc and δμ are normalized by the superfluid phase transition temperature Tenvc0 at δμ=0.

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    • Figure 9
      Figure 9

      (a1) Calculated intensity −Re[ΓR(q=(qx,qy,0),ν=2μ)] of the real part of the retarded particle-particle scattering at the solid square in (a2), in the absence of optical lattice. (a3) Positions of two edges imprinted on the momentum distribution of Fermi atoms by the two reservoirs, at |pFα|=2mμα=2m[μ±δμ]. |Q| in (a1) is just related to the size difference between the two edge circles shown in (a3). In calculating (a1)–(a3), we set (pFas)−1=−0.6 and γ/ɛF=0.02. Panels (b1)–(b3) show the case in the presence of the three-dimensional optical lattice: (b1) is the same plot as (a1) at the solid square in (b2). Panel (b3) is the same plot as (a3), determined from Eqs. (23) and (24). Qx+ in (b1) is related to the size difference between the two Fermi surface edges shown in (b3). We set U/(6t)=0.8, γ/(6t)=0.01, t′=0, and n=0.3 in (b1)–(b3).

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    • Figure 10
      Figure 10

      Calculated Tenvc (upper panels) and |QFF| (lower panels) in a driven-dissipative ultracold lattice Fermi gas, as functions of the chemical potential bias δμ and the next nearest neighbor hopping t′. (a) NMF theory. (b) NNSR theory. The solid circle is the boundary between the NBCS (solid line) and NFFLO (dashed line) phase transitions, which is also referred to as the Lifshitz point in the literature. In the NMF case, the temperature TenvLP at the Lifshitz point is always located at TenvLP/Tenvc0≃0.45, irrespective of the value of t′ (at least within the parameter region shown in this figure). In contrast, TenvLP decreases with increasing t′ in the NNSR case. We set U/(6t)=0.8, γ/(6t)=0.015, and n=0.3. Tenvc is normalized by the value at δμ=0 (≡Tenvc0).

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    • Figure 11
      Figure 11

      Positions of Fermi surface edges (“FSL” and “FSR”) produced by the two reservoirs for various values of the next-nearest-neighbor hopping t′. We take U=0, n=0.3, δμ/(6t)=0.1, γ→+0, and kz=0.

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    • Figure 12
      Figure 12

      Calculated Tenvc by the NNSR theory for various values of the filling fraction n. We set U/(6t)=0.8, γ/(6t)=0.01, and t′=0. Tenvc is normalized by the value at δμ=0 (≡Tenvc0).

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    • Figure 13
      Figure 13

      (a) Phase diagram of a driven-dissipative lattice Fermi gas, with respect to the environmental temperature Tenv, chemical potential bias δμ, and fictitious magnetic field h. The solid (dashed) line denotes the NBCS (NFFLO) phase-transition temperature Tenvc. δμc(h) and hc(δμ) are, respectively, the critical chemical potential bias and the critical magnetic field, above which the superfluid phase vanishes. The system is in the thermal equilibrium state at δμ=0, where the thermal equilibrium BCS and FFLO states are realized, depending on the magnitude of h. (b) The phase diagram at Tenv=0. We set n=0.3, t′=0, and γ→+0, and the NMF theory is used.

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    • Figure 14
      Figure 14

      (a) (a1) Positions of four edges imprinted on the momentum distribution of Fermi atoms: FS↑L (solid line), FS↑R (dashed line), FS↓L (dotted line), and FS↓R (dashed-dotted line), at the phase boundary (A4) in Fig. 13. These lines are obtained from Eqs. (23) and (24) at kz=0. (a2) Nesting vector Q1,x+ between the Fermi surfaces FS↑R and FS↓L. (a3) Nesting vector Q2,x+ between the Fermi surfaces FS↓R and FS↑L. Because of the fourfold symmetry of the background optical lattice, physically equivalent nesting vectors to Q1,x+ and Q2,x+ also exist in the −x direction, as well as the ±y and ±z directions. (b) Inverse retarded particle-particle scattering matrix −Re[ΓR(q=(qx,0,0),ν=2μ)−1], as a function of qx. Each result is at the phase boundary (A1)–(A4) in Fig. 13. When h≠0, −Re[ΓR(q,ν=2μ)−1] has two minima at the nesting vectors. As an example, we show the positions of |Q1,x+| and |Q2,x+| (>|Q1,x+|) in panel (b), where Q1,x+ and Q2,x+ are given in panels (a2) and (a3), respectively.

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    • Figure 15
      Figure 15

      (a) Calculated Tenvc when h/(6t)=0.175, as a function of the chemical potential bias δμ. (b) Fermi surfaces FSσ=↑,↓α=L,R (kz=0) at (b1)–(b4) in panel (a). In panel (b1), because δμ=0, the two Fermi surfaces FS↑=FS↑L,R and FS↓=FS↓L,R only exist.

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    • Figure 16
      Figure 16

      (a) Calculated Tenvc as a function of δμ in the high magnetic-field regime [h/(6t)=0.25], where the superfluid phase no longer exists in the thermal equilibrium state (δμ=0). (b) Fermi surfaces FSσ=↑,↓α=L,R (kz=0) at (b1) and (b2) in panel (a).

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