With tunable atom–atom interaction through Feshbach resonance, ultracold Fermi gas has provided an ideal platform to study the physical properties in the crossover from Bardeen–Cooper–Schrieffer (BCS) superfluid with weak attraction to a Bose–Einstein condensation (BEC) of bound pairs with strong attraction.^[1] In recent years, spin–orbit coupling (SOC), which is a key ingredient for many interesting quantum phenomena like topological insulators and quantum spin Hall effects in condensed-matter systems for electrons,^[2,3] has been realized experimentally by controlling atom–light interaction in ultracold atoms.^[4–7] Motivated by this new progress, the effects of SOC on the superfluid in ultracold atoms have become a cutting-edge field due to clean environment and highly tunable parameters. As SOC breaks the inversion symmetry and significantly changes the Fermi surface, the SOC Fermi atom gas with effective Zeeman fields^[8] can exhibit many intriguing phenomena,^[9–14] such as topological superfluids, Majorana modes, spin textures, and skyrmions. On the other hand, the polarized Fermi gas has been extensively investigated both experimentally and theoretically in the past decade.^[15–17] Due to the competition between the Cooper pairing and the mismatch of Fermi surfaces with different spin species, various exotic phases have been proposed in polarized Fermi gas.

An essential signature of rotating atomic superfluid in a trapping potential is the appearance of quantized vortices.^[18–20] Such vortex excitations have been observed for both population-balanced^[21] and population-imbalanced^[22] systems. The vortex cores consist of rotating normal atoms with quantized angular momenta. Since the effect of the Coriolis force acting on a neutral atom in a rotating system is similar to that of the Lorentz force acting on a charged particle in a uniform magnetic field, rotating atomic systems may exhibit integer and fractional quantum Hall physics.^[23–26]

In the absence of SOC, the population-balanced Fermi gas with adiabatic rotation has been studied by using the quantum Monte Carlo method together with the local density approximation (LDA) when vortices are not excited.^[27,28] The BCS–BEC evolution in this kind of system has been investigated with the BCS mean-field approximation^[29] and the Bogoliubov–de Gennes (BdG) approach.^[30] It was shown that adiabatic rotation, which breaks some of the superfluid pairs via the Coriolis force, could lead to a phase separation between the non-rotating superfluid at the center of trapping potential and the normal gas at the edge. In addition, the rotation-induced pair-breaking mechanism in the population-balanced Fermi gas with SOC was analyzed via the mean-field approximation in the entire BCS–BEC evolution.^[31]

In this paper, we study the polarized Fermi gas with Rashba SOC and adiabatic rotation near a wide Feshbach resonance on the ground state for a two-dimensional (2D) case. Adiabaticity requires that the rotation is introduced slowly to the system. Based on the BCS mean-field method with LDA, we obtain the analytic energy spectrum in this system. Due to the existence of metastable solutions to the gap equation that are typical in the population-imbalanced systems,^[32] we minimize the thermodynamic potential to ensure the order parameter solution is stable. The dependence of the minimum of thermodynamicpotential on the SOC strength and the rotation velocity are investigated. Besides, we calculate the condition for the system to exhibit topologically non-trivial phase, in which the zero-energy Majorana fermions in the edge state could be observed. Finally, the effect of adiabatic rotation on the distributions of number density are present by numerical calculation. Here, we ignore the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state, which is still hard to observe in experiment, and focus on the region of BCS pairing.

The paper is organized as follows. In Section 2, the model and the calculation method are introduced. In Section 3, we analyze the results obtained in Section 2 and discuss the energy spectrum and topological condition. The distribution of number density is also shown in this section. In Section 4, we summarize the study and give the conclusions.

To study the interplay between Rashba spin–orbit coupling and adiabatic rotation in a 2D polarized Fermi gas, we begin with the Hamiltonian density in the rotating frame^[13,33] 1 where μ_σ is the chemical potential for atoms with spin σ = {↑,↓}, N_σ denotes the total number of particles with spin σ. H₀(r) is the unperturbed term with adiabatic rotation in a harmonic confinement potential, H_soc is the SOC interaction, and H_int is the s-wave interaction between the two fermionic species. They are 2 H0(r)=∑k,σ(εk−μσ+V(r)−Ωrot⋅L)ak,σ†ak,σ,  3 4 where the kinetic energy ε_k = ħ²k²/(2m), V(r) = mω²r²/2 is the harmonic potential. Here we assume that the harmonic potential varies slowly in space and the number density in the center of the trapping is sufficiently large, so the LDA is valid. Ω_rot = Ω_rote_z is the angular frequency and L = r × p is the orbital angular momentum. (a_k, σ) denotes the creation (annihilation) operator for a fermion with momentum k and spin σ. α is the strength of Rashba spin–orbit coupling which can be tuned by changing the parameters of the gauge-field generating lasers,^[34] φ_k = arg(k_x + ik_y), U is the bare s-wave contact interaction strength and can be renormalized by the standard relation in 2D system^[35] 5 1U=∑k12εk+Eb. Here, E_b > 0 is the binding energy of the two-body bound state in two dimensions without SOC. As E_b increases from zero to a large value, the system undergoes BCS to BEC. Therefore, the variation of E_b can be used to represent the BCS–BEC crossover.

Taking the pairing function Δ = UΣ_k〈a_−k,↓a_k,↑〉, one gets the effective Hamiltonian in the Nambu basis as 6 H−∑σμσNσ=12∑kΨ†(k)ℋ(k)Ψ(k)+∑k(εk−μr−Ωrot⋅L)+|Δ|2U with 7 ℋ(k)=(H˜0(k)ΔΔ*−σyH˜0∗(−k)σy) 8 Here, σ_i (i = x, y, z) are the Pauli matrices, ξ_k = ε_k − μ_r, and the local chemical potential μ_r = μ − V(r) with μ = (μ_↑ + μ_↓)/2. h is defined as h = (μ_↑ − μ_↓)/2. Both μ and h can be tuned by changing the population in the two different hyperfine spin states. As the Hamiltonian is symmetric with respect to a spin flip, the Physical properties would be the same regardless of which spin species is the majority component. Without loss of generality, we take h > 0 in the paper.

The Hamiltonian is quadratic and can be diagonalized as 9 H(r)−∑σμσNσ=​​​​​​​∑k,s=±Ek,sbk,s†bk,s+|Δ|2U+12∑k,s=±(ξk,s−Ek,s−Ωrot⋅L),  where (b_k,s) is the creation (annihilation) operator for the quasiparticles with the excitation spectra 10 where In contrast to the non-rotating case,^[36] here E_k,± are not positive semi-definite, but rather dependent on actual physical parameters. When the energy of quasi-particle excitations E_k,s(s = ±) become negative, the vacuum of quasiparticles are not the ground state anymore, and one needs to fill up the negative energy modes to construct the lowest-energy ground state.^[32,37] As a result, the thermodynamic potential Ω is obtained as 11 Ω=−1β∑k,sln(1+e−β|Ek,s|)+∑k,sθ(−Ek,s)Ek,s​​​​+12∑k,s=±(ξk,s−Ek,s)+|Δ|2U, where β = 1/k_BT and k_B is the Boltzmann constant. The pairing gap should be determined by minimizing the thermodynamic potential ∂Ω/∂Δ = 0, and the particle number equations can be given by n_σ(r) = −∂Ω/∂_μσ(r). They are derived as 12 |Δ|∑k[∑s=±[f(Ek,s)−12](1+sh2E0)Ek,s+Ωrot⋅L+22εk+Eb]=0  and 13 nσ(r)=∑k{12−∑s=±(f(Ek,s)−12)[−(εk−μ)+δσh2(Ek,s+Ωrot⋅L)+s−(εk−μ)(α2k2+h2)2(Ek,s+Ωrot⋅L)E0+sδσh((εk−μ)2+|Δ|2)2(Ek,s+Ωrot⋅L)E0]}, where f(x) = 1/[1 + exp(βx)] is the Fermi distribution function and δ_↑ = −δ_↓ = 1. Here, it is worth emphasizing that the gap equation given by Eq. (12) may have unstable solutions in certain parameter regions that are typical in the population-imbalanced systems.^[32] To overcome this problem, we will always check the stability of the solution by finding out all the minima of the thermodynamical potential Ω. For convenience, we take Δ to be real, namely Δ = |Δ|. The total particle number N = N_↑ + N_↓ and N_σ is the total particle number for each spin species which can be determined from a trap integration: N_σ = ∫ d²rn_σ(r). Here we are interested in the low temperature where the condensate is quite large, so we restrict our study to the zero temperature in the following discussions.

To make the result universal and applicable to the system, we take the Fermi energy E_F as the unit of energy for N noninteracting fermions, where .^[14,31] The dimensionless harmonic trapping potential can be expressed as V(r)/E_F = r²/R², where R is the Thomas–Fermi radius in the 2D case and is given by . The rotation velocity in the dimensionless form is v = Ω_rotr/(ωR). In Fig. 1, the thermodynamic potential is plotted as a function of Δ for various rotation velocities v and spin–orbit coupling strength αk_F/E_F. Here k_F is the Fermi momentum and is defined as . One sees that the minimum of the thermodynamic potential shifts to the lower value of Δ when rotation velocity v increases, as shown in Fig. 1(a). If the rotation velocity v is large enough, the minimum of the thermodynamic potential will appear at Δ = 0, which means that the superfluidity is totally destroyed. On the other hand, as the spin–orbit coupling strength α increases, the minimum of the thermodynamic potential shifts to the higher value of Δ as shown in Fig. 1(b).

Fig. 1.

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	Fig. 1. (color online) Thermodynamic potential as a function of the pairing order parameter Δ with various rotation velocity v(a) and SOC strength αkF/EF (b). The two-body binding energy is chosen as Eb/EF = 0.5. The local chemical potential μr/EF = 0.6. h/EF = 0.5. Other parameters are given as: (a) αkF/EF = 0.1; (b) v = 0.325.

Due to the presence of adiabatic rotation, the excitation energy for the branches with E_k,+ and E_k,− in Eq. (10) may become zero even when the order parameters are nonvanishing. Consider the condition for E_k,± = 0, namely, 14 When equation (14) have real solutions in momentum space, the excitation gap is zero in the presence of a non-vanishing pairing gap. This case corresponds to the so-called breached pair state.^[38,39] Here, both the E_k,+ and E_k,− branches can become zero with the corresponding momenta. This is different from the case of the polarized Fermi gas without SOC and rotation,^[37] in which only E_k,− branch could have gapless excitations. In Fig. 2, we plot the contour lines of the excitation spectra E_k,± with various rotation velocity v in the two-dimensional k plane. One can see that the gapless contour begins to appear when rotation velocity reaches a critical value. As rotation velocity continues to increase, the gapless contour becomes larger.

Fig. 2.

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	Fig. 2. (color online) The contour lines of the excitation spectra Ek,+ (left) and Ek,− (right) in kx-ky plane. The local chemical potential μr/EF = 0.8, the SOC strength αkF/EF = 0.8, h/EF = 0.36, Eb = 0.41EF. Other parameters are given as: For panels (a1) and (b1), v = 0.3; For panels (a2) and (b2), v = 0.35; For panels (a3) and (b3), v = 0.45; For panels(a4) and (b4), v = 0.5.

The BdG Hamiltonian ℋ(k) in Eq. (7) satisfies the particle–hole symmetry Ξℋ(k)Ξ⁻¹ = −ℋ(−k), where and Λ = iσ_y ⊗ τ_y, τ_y is the Pauli matrix. ! is the complex conjugate operator. The parameter region for the topological superfluid is determined by the topological index ℳ = sign(Pf{Γ}),^[40] where Pf{Γ} represents the Pfaffian of the skew matrix Γ = ℋ(0)Λ. When ℳ = −1, the system shows a topological superfluid phase, while a conventional superfluid phase is shown with ℳ = 1. The topologically non-trivial phase is realized when 15 The first condition in Eq. (15) requires that . Here E_gap = min(E_k,s) defines the bulk quasi-particle excitation gap of the system with excitation spectra E_k,s. The condition min(E_k,s) > 0 ensures the bulk quasi-particle excitations are gapped to protect the zero-energy Majorana fermions in the edge state with the topological regime. The transition from topologically trivial phase to topologically non-trivial phase can be better understood by observing the closing and reopening of the energy gap E_gap, which is necessary to change the topological structure of Fermi surface. In Fig. 3(a), we plot the quasi-particle excitation spectrum E_k,− and quasi-hole excitation spectrum determined by the Hamiltonian in Eq. (7). One sees that the excitation spectra satisfy the particle–hole symmetry. Besides, with h changing from 0.37E_F to 0.54E_F, the topological structure of the Fermi surface is changed. In order to better show the changes, we plot E_gap as a function of h in Fig. 3(b). It can be directly seen that the topological structure of the Fermi surface is changed with the closing and reopening of E_gap near the topological transition .

Fig. 3.

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	Fig. 3. (color online) (a) The quasi-particle excitation spectrum Ek,− (solid lines) and quasi-hole excitation spectrum (dashed lines) with kx = 0. For the green lines, h = 0.37EF; for the red lines, h = 0.51EF; for the blue lines, h = 0.54EF. (b) The energy gap Egap as a function of h. Other parameters are: μr/EF = 0.2, αkF/EF = 0.7, v = 0.053, Eb = 0.35EF.

Fig. 4.

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	Fig. 4. (color online) The distributions of number density ñ↑ (a), ñ↓ (b), δñ = ñ↑−ñ↓ (c), and order parameter Δ/EF (d) as functions of the dimensionless distance from the trap center with angular frequency Ωrot = 0 and 0.33ω. The parameters are Eb = 0.35EF, αkF/EF = 0.7, h = 0.3EF.

Next, we study the distribution of number densities and order parameter with adiabatic rotation. The number equation in a dimensionless form can be written as 16 where is dimensionless number density, and n_σ is the number density given by Eq. (13) at zero temperature. The dimensionless position . If the parameters E_b, α, h are fixed, the properties of the system only depend on the angular frequency Ω_rot. In Fig. 4, we plot ñ_↑ (a), ñ_↓ (b), δñ = ñ_↑ − ñ_↓ (c), and order parameter Δ/E_F (d) as functions of the dimensionless distance from the trap center with angular frequency Ω_rot = 0 and 0.33ω. One sees that due to the presence of adiabatic rotation, the polarized Fermi gas is separated into two different phases from the trap center to the edge. One of them is the gapped superfluid phase occupying the center of the trap (r < r_c), the other is the gapless superfluid phase out from the center (r > r_c). Here r_c represents the position where gapless superfluid phase begins to appear. For the gapped superfluid phase, the number density and order parameter have nothing to do with adiabatic rotation and the physical quantities remain the same as the Ω_rot = 0. However, in the gapless superfluid phase, the number density ñ_↑ is increased while the ñ_↓ is decreased by the rotation. This opposite effect leads to the enhancement of the polarization of the Fermi gas.

In the paper, we have developed a mean-field theory to characterize the superfluid properties of two-dimensional polarized Fermi gas with SOC and adiabatic rotation in a harmonic trapping potential. Under LDA, we have obtained the analytic energy spectra of this system and studied the dependence of energy spectra on rotation velocity by numerical calculation. It is found that both the two quasi-particle excitation spectra may become zero even when the order parameters are nonvanishing, which is different from the case of polarized Fermi gas without SOC and rotation that only one branch could have gapless excitations.^[37] Furthermore, we have investigated the conditions for the topologically non-trivial superfluid phase, in which the zero-energy Majorana fermions in the edge state could be observed. Finally, we study the effects of adiabatic rotation on the number density with different spins and find that the polarization of Fermi gas could be enhanced by rotation.