Mesoscopic spin transport between strongly interacting Fermi gases

We investigate a mesoscopic spin current for strongly interacting Fermi gases through a quantum point contact. Under the situation where spin polarizations in left and right reservoirs are same in magnitude but opposite in sign, we calculate the contribution of quasiparticles to the current by means of the linear response theory and many-body T -matrix approximation. For a small spin-bias regime, the current in the vicinity of the superfluid transition temperature is strongly suppressed due to the formation of pseudogaps. For a large spin-bias regime where the gases become highly polarized, on the other hand, the current is affected by the enhancement of a minority density of states due to Fermi polarons. We also discuss the broadening of a quasiparticle peak associated with an attractive polaron at a large momentum, which is relevant to the enhancement.


I. INTRODUCTION
Quantum simulation with ultracold atomic gases allows one to explore regimes of quantum-many body problems where conventional systems such as condensed matter and nuclear matter are hard to reach [1][2][3]. A strongly-interacting Fermi gas realized with the Feshbach resonance is the prototype example, and revealed existences of Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover and the unitarity regime in which the typical length scale characterizing the atomic interaction disappears [4][5][6]. Whilst both theoretical and experimental progresses have deepened understanding of the bulk phase structure, understanding of the non-equilibrium properties remains challenging.
Recently, quantum transport of strongly-interacting Fermi gases attracts rising attention in association with the atomtronics devices where non-equilibrium properties on circuit or two-terminal systems are investigated [7]. By using the controllable ultracold Fermi gases, quantum point contact [8], and junction systems attached with two [9] or three [10] dimensional reservoirs have experimentally been implemented. In the case of the stronglyinteracting superfluid regime, superfluid transport such as nonlinear current-bias characteristics induced by the multiple Andreev reflections [11], and the AC Josephson oscillation [10] and DC Josephson effect [12] has been confirmed. In the case of the normal Fermi gas with a strong attractive interaction, a conductance beyond the quantized value has been found [13].
In addition to controllability of the interaction, transport systems with ultracold Fermi gases have an advantage that spin transport can directly be measured with a tunable spin bias [13]. We note that this is in contrast to condensed matter systems where spin transport is normally measured in an indirect manner [14]. In the context of the interacting Fermi gases, while mass transport * These two authors contributed equally to this work. contains information both on single-particle and collective excitations, spin transport is not involved in Cooper pair transport [15,16]. Thus, it is expected that the spin transport measurement becomes a sensitive probe of single-particle excitations.
In this paper, we investigate mesoscopic spin transport in attractively interacting two-component Fermi gases. We consider the tunneling regime of a two terminal system consisting of two normal Fermi gases through a quantum point contact (see Fig. 1). Spin polarizations of the Fermi gases in left and right reservoirs are assumed to be equal in magnitude but opposite in sign and are controlled by the parameter h. Due to this spin imbalance, spin-up and spin-down fermions can move in opposite directions, and therefore a nonzero spin current without flow of a mass current is generated. We focus on the strongly interacting regime near unitarity, where the absolute value of a s-wave scattering length |a| is much larger than the interatomic distance, and intercomponent fermions strongly interact with each other. Figure 2 is the phase diagram of the spin-imbalanced Fermi gases at unitarity [17,18]. When the low temperature and small h regime is concerned, each reservoir becomes the superfluid where spin excitations are expected to be frozen. Therefore, in this work, we elucidate spin transport above the superfluid critical temperature.
In contrast, excitations of a highly polarized Fermi gas realized in the large h regime is governed by Fermi polarons, which are mobile impurities immersed in a Fermi sea [49]. Polaronic properties such as renormalization factors, effective masses, and polaron energies have been measured with RF spectroscopy [50][51][52][53][54][55][56][57]. Correspondingly, a bunch of theoretical works have been made [58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77], most of which consider the zero-temperature and single-polaron limit by assuming that the impurity chemical potential is equal to the polaron energy. However, we note that the theory based on such an ideal limit is not available to discuss polaronic effects in spin transport at finite temperature and given chemical potential. To incorporate the polaronic properties in a correct fashion, we perform the finite-temperature many-body formalism of Fermi polarons. By using this formalism, moreover, we show that the crossover from the pseudogap to the polarons can be explored through spin transport.
This paper is organized as follows. In Sec. II, we present the formalism of the tunneling Hamiltonian approach together with the diagrammatic T -matrix approximation. Section III is devoted to discussing how excitation properties in strongly interacting Fermi gases in reservoirs affect spin transport. We conclude this paper in Sec. IV.
Throughout this paper, we set = k B = 1 and the volumes for both reservoirs are taken to be unity.

II. FORMALISM
In order to study spin transport of two terminal systems in normal Fermi gases with strong interaction, we begin with the following grand canonical Hamiltonian: where left and right reservoirs are referred to as j = L, R, respectively, c p,σ,j is the annihilation operator of a fermionic atom with spin σ = ↑, ↓ in the reservoir j, and ξ p,σ,j = p 2 /2m − µ σ,j is a single-particle energy measured from the chemical potential µ σ,j . The interaction between fermions in the reservoir j is attractive (U > 0) and related to the s-wave scattering length a by 1/U = p m/p 2 − m/(4πa). The term in H T denotes the tunneling of fermions from one reservoir to the other and t characterizes the strength of the tunneling. As mentioned above, we focus on the situation where there is a pure spin bias shown in Fig. 1. The majority and minority components in the left reservoir are ↑ and ↓, respectively, and those in the right reservoirs are the other way around. It follows that the parameter h controls not only the polarizations in both reservoirs but also a bias in spin transport. We assume that both reservoir has same temperature T , which is above the superfluid transition temperature. For convenience, we define a new label α ≡ (σ, j) and, hereafter, the majority components (σ, j) = (↑, L), (↓, R) are referred to as "α = +," while the minority ones (σ, j) = (↓, L), (↑, R) as "α = −." The chemical potentials of the majority (α = +) and minority (α = −) are µ ± = µ ± h as depicted in Fig. 1.
The spin current operator in the Heisenberg representation is given byÎ p,σ,j c p,σ,j is the particle number operator. By using the linear response theory, the spin current to leading order in the tunneling amplitude t can be obtained in a similar way as for a mass current [78]. For a steady state, the spin current is obtained as (see Appendix A) where ρ α (ω) is the DOS for the majority (α = +) [minority (α = −)] and f (ω) = 1/(e ω/T + 1) is the Fermi distribution function. We note that regardless of value of h, Eq. (2) is correct up to t 2 . In order to take contributions of pair correlations to ρ α (ω) into account, we employ the extended T -matrix approximation (ETMA) [17,71]. The density of states is related to an analytically-continued Matsubara Green's function G α (p, iω n ) with ω n = (2n + 1)π T (n ∈ Z) as follows: where ξ p,α = p 2 /2m − µ α , and δ is an infinitesimal positive constant. Within the ETMA, the self energy Σ α (p, iω n ) and the many-body T -matrix Γ(q, iν ) with ν = 2 π T ( ∈ Z) are given by Feynman diagrams in Fig. 3, leading We note that the self-consistent programme above can reduce unphysical results. For example, the ordinary Tmatrix approximation is known to suffer from a negative spin susceptibility in the strong-coupling regime. However, ETMA spin susceptibility takes a positive value in the whole crossover regime [17]. We now discuss the choice of parameters (a, T, µ, h) in this work. In an unpolarized case (h = 0), we fix (a, T, µ) as follows. For a given density n 0 of the total particle number in each reservoir, the corresponding Fermi momentum and temperature are provided by k F,0 = (3π 2 n 0 ) 1/3 and T F,0 = k 2 F,0 /(2m), respectively, and two dimensionless parameters (k F,0 a) −1 and T /T F,0 are fixed. Then, (k F,0 a) −1 → −∞ corresponds to the weak interaction limit and (k F,0 a) −1 → +∞ to the strong interaction limit in the fermion language. The spinaveraged chemical potential µ is determined so that the particle number equation in the absence of h, is satisfied. Then, the fictitious Zeeman field h is changed with (a, T, µ) fixed. Since we are interested in how strong correlations affect spin transport, we consider the regime |k F,0 a| −1 1 near unitarity. Figure 2 shows the phase diagram of the Fermi gases at unitarity ((k F,0 a) −1 = 0) in the (h, T )-plane. The transition temperature is determined as the temperature satisfying the Thouless criterion [79] given by [Γ(0, 0)] −1 = 0. We note that at low temperature there is the first-order phase transition between normal and superfluid phases, including the socalled Fulde-Ferrel-Larkin-Ovchinnikov state [80]. To address these transitions, one has to calculate the thermodynamic potential in each phase, which is out of scope in this paper. Figure 4 shows the number densities n ± of majority and minority atoms at unitarity as functions of h. (Note that the total number density n + + n − changes as h increases.) The monotonic behavior of x = n − /n + in Fig. 4 means that, as h becomes larger, the gases in both reservoirs become more highly polarized. As shown in Figs. 10 and 11 in the next section, the polaron picture becomes valid for the large h regime.

III. RESULT
We now discuss spin transport properties for normal Fermi gases with strong interparticle interactions under the configuration shown in Fig. 1. Figure 5 shows the spin conductance G spin ≡ I spin /(2h) at unitarity. We can see that G spin grows with increasing h with a fixed T . Furthermore, it is remarkable that G spin is largely suppressed in the low-T and low-h regime, where the pseudogap emerges. As shown in Fig. 6, our ETMA calculation for unpolarized (h = 0) and polarized (h = 0.25ε F,0 ) gases at T = T c can confirm the pseudogap structures of DOSs, being the signature of the preformed Cooper pairs in the BCS-BEC crossover regime of an ultracold Fermi gas.
To make the effect of the pseudogap on spin transport clearer, let us focus on the zero-bias limit. In this limit, majority and minority DOSs become identical,ρ(ω) = ρ ± (ω)| h=0 , and the spin conductance reduces to the following form: limit. We note that the calculation of G spin is stopped at the transition temperature T c . Away from T c , G spin increases with decreasing T because of quantum degeneracy of fermions. On the other hand, as shown in Fig. 6, the DOS has a dip structure around ω = 0 near the superfluid transition. Since −∂f (ω)/(∂ω) ∝ cosh −2 (ω/2T ), the spin conductance is sensitive toρ(ω) around ω = 0. Therefore, the appearance of this pseudogap leads to a large suppression of G spin . The single-particle excitation is strongly suppressed due to the formation of spin-singlet pairs in the pseudogap regime. This suppression leads to the so-called spin gap in the temperature dependence of the spin susceptibility [20,[34][35][36][37]39]. We note that the spin-gap temperature, where the spin susceptibility starts to be suppressed due to strong pairing fluctuations, is T SG = 0.37T F,0 at unitarity [39]. Although it is the crossover temperature and there are ambiguities for the definition to characterize these phenomena, the maximum temperature of G spin at unitarity is also close to T SG . This result indicates that G spin is also useful to study pseudogap physics. From Fig. 7, we can also see the interaction dependence of G spin . In the weak-coupling side ((k F,0 a) −1 = −0.5 in Fig. 7), G spin becomes larger compared to that at unitarity. However, even at this coupling, G spin near T c decreases due to the pairing fluctuation effects as decreasing temperature. As in the case of the spin susceptibility [39], the spin conductance is strongly suppressed as increasing strength of the pairing interaction. At stronger couplings, the gases in both reservoirs are dominated by tightly bound molecules and the spin degree of freedom tends to be frozen. In this case, only the thermally dissociated atoms contribute to the spin susceptibility as well as spin transport.  Figure 8 represents a crossover of the spin current at unitarity from the pseudogap regime to the polaronic regime by shifting h. We can see that I spin for small h is smaller than in the non-interacting counterpart, where the current is analytically given by Eq. (B2) in Appendix B. As explained in the discussion of G spin , this suppression is caused by the pseudogap in the region where h is small. Figure 9 shows the calculated DOSs for both majority and minority components at T = 0.21T F,0 for various h. When h becomes larger, polarizations of the gases in both reservoirs grow and the pseudogap structures of ρ ± (ω) vanish since the gases go away from T c at a fixed temperature. The majority DOS is enhanced in the whole energy region with increasing h due to the increase of n + . In the large-h region, ρ + (ω) coincides with a DOS in an ideal Fermi gas given by Eq. (B1), since under a large population imbalance, minority atoms cause a negligible effect to a large number of majority atoms. On the other hand, the minority DOS shows a more complex modification than ρ + (ω) with increasing h. In particular, in the large-h regime, minority atoms can be regarded as the so-called Fermi polarons. In our configuration of chemical potentials with fixed µ = (µ + + µ − )/2, I spin is enhanced compared with that without self-energy corrections [81]. This implies that the polaronic quasi-particle excitations encoded in the self-energy corrections play an important role in spin transport for a large spin bias.
In order to discuss the contributions of the polaronic transport to I spin for large h, we start with the investigation of the single-particle spectral functions defined by In the literature on the Fermi polarons, the Fermi momentum k F,+ = (6π 2 n + ) 1/3 and the Fermi energy ε F,+ = k 2 F,+ /(2m) for majority atoms are conventionally taken as units of energy and momentum. Thus we use these units to discuss A α (p, ω) in the large spin-bias regime. For large h, the majority spectral function can be replaced by that for free fermions A + (p, ω) = δ(ω +µ + − p 2 2m ). On the other hand, Fig. 10 shows that the minority spectral functions near unitarity are distinct from its non-interacting counterpart. In particular, there are two types of characteristic excitations: a sharp peak at low energy corresponding to an attractive polaron and a broad peak around ω + µ − ≈ ε F,+ associated with a repulsive branch or repulsive polarons. Since A − (p, ω) at low energy is dominated by the attractive polaron, it is well approximated by where Z a , m * a , and E a are the renormalization factor, the effective mass, and the energy of the attractive polaron, respectively. By integrating this approximated form over p, the contribution from the polaron to ρ − (ω) is found to be for ω > −µ − + E a . Figure 11 shows the minority DOS at unitarity numerically calculated in the ETMA. The obtained ρ − (ω) for ω + µ − ε F,+ is found to be enhanced compared to a DOS without self-energy corrections. Fitting the calculated DOS at low energy by using Eq. (11), we obtain Z a (m * a /m) 3/2 = 0.945 and E a = −0.627ε F,+ . These results are in good agreement with the zerotemperature results in the single-polaron limit, where Z a = 0.78, m * a /m = 1.17, and E a = −0.606ε F,+ , leading to Z a (m * a /m) 3/2 = 0.987 [59,64]. This means that the effects of finite T and x on ρ − (ω) are not so important in this temperature and bias range [71]. For 0 ω + µ − ε F,+ , ρ − (ω) deviates from Eq. (11). In this energy range, the enhancement of ρ − (ω) comes from not only the attractive polaron at high momentum, whose lifetime is finite, but also the repulsive branch. In particular, the broadening of the peak associated with the attractive polaron in Fig. 10 can intuitively be understood as follows. The polaron is a quasiparticle consisting of a minority atom surrounded by majority atoms. This quasiparticle picture is valid when the velocity of a dressed minority atom v − ≈ p/m * a is smaller than a typical velocity of the majority atoms v F,+ = k F,+ /m. When v − > ∼ v F,+ , the majority atoms can no longer follow the fast-moving minority atom and thus the attractive polaron tends to be unstable. At unitarity, this unstable regime is estimated as p > ∼ (m * a /m)k F,+ ≈ 1.17k F,+ and is consistent with the region where the peak associated with the attractive polaron becomes broad (see Fig. 10). This mechanism of the broadening is analogous to the Cherenkov instability in Bose polarons [82,83], where a minority atom undergoes the supersonic regime associated with the Bogoliubov phonons. Actually, this intermediate energy range of polaron spectra plays a significant role to understand polaronic spin transport as discussed below. For sufficiently large ω, ρ − (ω) becomes close to its non-interacting counterpart, which is consistent with the asymptotic behavior derived by the operator product expansion [84].
Let us now come back to I spin for a large spin bias. Figure 12(a) shows functions in the integrand of Eq. (2) at T = 0.25T F,0 and h = ε F,0 . We can see that ρ + (ω − h) is almost consistent with that without self-energy corrections. Because of the existences of f (ω − h) − f (ω + h) reflecting the Fermi-Dirac statistics and of a threshold energy ω = −µ for ρ + (ω − h), ρ − (ω + h) contributes to I spin only over the region where the attractive polaron at finite momentum and the repulsive branch appear. We note that in the absence of the pairing interaction, both curves coincide with each other. Thus, the enhancement of the spin current in the highly polarized regime originates from the polaron excitations in the intermediateenergy range (0 ω + µ − ε F,+ in Fig. 11). Figure 12(b) compares I spin with and without selfenergy corrections at (k F,0 a) −1 = −0.5, 0, and 0.5. While the enhancement of I spin is large in the weak-coupling side, it is small in the strong-coupling side. Since the magnitude of I spin depends on µ obtained by solving Eq. (7) at each (k F,0 a) −1 in our configuration, I spin becomes larger in the weak-coupling side where µ is positively large compared to that in the strong-coupling side. In this sense, the polaron properties appear as the ratio given by I spin with and without the self-energy corrections. This ratio is still larger in the weak-coupling side. The enhancement of I spin is associated with the overlap of ρ + (ω + h) and ρ − (ω − h) around ω = 0 shown in Fig. 12(a). While these two DOSs in the unitary limit as well as the weak-coupling side have a relatively large overlap due to the small polaron energy, such an overlap becomes smaller in the strong-coupling side due to the large polaron energy. Physically, the large polaron energy indicates the strong binding of Fermi polarons.
Since there is no single-particle state of majority atoms in the energy range corresponding to low-energy attractive polarons, these polaronic states are irrelevant to spin transport. While the minority DOS at (k F,0 a) −1 = 0.5 is enhanced in the low-energy region (ω < 0) due to the polaron binding effect, that around the energy region where majority DOS ρ + (ω + h) starts to be finite (ω > ∼ 0.2ε F,+ ) is relatively insensitive to the interaction. In addition, the strong attraction makes Z a of the attractive polaron smaller compared with that at unitarity [64], which is expected to reduce the enhancement of spin transport. Therefore, one can find that the non-equilibrium spin transport in highly polarized regime is enhanced by the broadened polaron spectra in the intermediate energy region, and suppressed by the polaron binding effect.

IV. CONCLUSION
In this paper, we elucidated mesoscopic transport properties of spins for strongly interacting Fermi gases connected via a quantum point contact. The tunneling Hamiltonian formalism was used to investigate a steady spin current between two spin-polarized Fermi gases. By employing the linear response theory combined with the diagrammatic approach, the spin current I spin and spin conductance G spin in the normal phase were computed for a wide range of parameters. We found that the emergence of the pseudogap results in a large suppression of spin transport in low-T and low-h regime as shown in Figs. 5-8. On the other hand,the gases become highly polarized for a large spin bias h. In this case, both the attractive polarons at finite momenta and the repulsive branch play significant roles and they lead to the enhancement of I spin compared with the current without self-energy corrections.
As mentioned in previous studies on the spin susceptibility for a strongly interacting Fermi gas [17,[34][35][36][37][39][40][41][42], spin properties are sensitive to the formation of a pseudogap. Mesoscopic spin transport with a small spin bias studied here provides a new probe to experimentally examine the pseudogap phenomenon. We also clarified that the spin current for a large spin bias is affected by excitations including both the attractive polarons at finite momenta and the repulsive branch. At the same time, a set of chemical potentials µ σ,j considered in this article (see Fig. 1) has limitation to examine the properties of polarons in the whole energy range. Such polaronic properties are considered to be accessible in a two terminal system under another choice of µ σ,j with the use of the spin filter, which has been recently realized in an ultracold atom experiment [85]. By assuming µ ↓,L = µ ↓,R µ ↑,L = µ ↑,R +V ↑ and filtering out the majority (σ =↓) component, the fully polarized current for a small bias V ↑ encodes information of attractive polarons at low energy.
Our method can be generalized to fermionic superfluids [86]. Unlike the mass current case, the Josephson current is expected not to contribute to the spin current. Another generalization is the study beyond the linear response theory to discuss the good-contact regime. While such an analysis is generically complicated, the quasiparticle current takes the form of Eq. (2) up to a prefactor in some situation (see Appendix C). Our formalism also predicts the noise of the spin current. Within the linear response theory, the noise at zero frequency is related to I spin by S(ω = 0) = coth(h/T )I spin as derived in Appendix A. We expect such a noise to be accessible in future ultracold atom experiments [87]. This appendix is devoted to the linear response theory to the tunneling amplitude t. For convenience, we introduce C σ ≡ t p,p c † p,σ,R c p ,σ,L andÎ σ (t ) ≡ −dN σ,L (t )/dt . The mass and spin current operators are given byÎ mass =Î ↑ +Î ↓ andÎ spin =Î ↑ −Î ↓ , respectively. The tunneling Hamiltonian in Eq. (1c) is rewritten as H T = σ=↑,↓ (C σ + C † σ ). Using the Heisenberg equation combined with {c p,σ,j , c † p ,σ ,j } = δ pp δ σσ δ jj and {c p,σ,j , c p ,σ ,j } = 0, we obtainÎ σ (t ) = i[N σ,L , H] = −iC σ + iC † σ . First, we review the linear response of the current for the spin-σ component. We follow the procedure in Ref. [78]. Hereafter, · · · denotes the expectation value for a given non-equilibrium state. According to the Kubo formula, I σ ≡ Î σ (t ) for a steady state is given by where · · · eq is a thermal average for the Fermi gases in both reservoirs, and O (A) (t ) ≡ e iAt Oe −iAt has been defined. Operators in the right hand side are in the Heisenberg representation for H 0 = K L + K R + σ,j µ σ,j N σ,j . Using the Baker Campbell Hausdorff formula, we can find C (H0) σ (t ) = e −i∆µσt C (K0) σ (t ), where K 0 = K L +K R and ∆µ σ = µ σL − µ σR . (Henceforth, the shorthand notation C σ (t ) = C (K0) σ (t ) is used only for the operator C σ .) By employing this as well as the expressions ofÎ σ and H T in terms of C σ , Eq. (A1) in the normal phase can be rewritten as where is a retarded correlation function. By using the commutability of K L and K R in Eq. (1) as well as an interrelation between retarded and Matsubara Green's functions, I spin can be rewritten as Eq. (2) in terms of DOSs.
Combining this with Eqs. (A2) and (A6), we obtain In the case of the chemical potentials shown in Fig. 1, we have ∆µ ↑ = −∆µ ↓ = 2h. Therefore, the noise is related to I spin by S(0) = coth(h/T )I spin .
In the end of this appendix, we will comment on the noise of the mass current, which is given by replacing δÎ spin in Eq. (A4) with δÎ mass =Î mass − Î mass . As mentioned above, cross correlation functions between currents with opposite spins vanish in the normal phase. As a result, both mass-and spin-current noises have the same form of Eq. (A6). When a bias is spin independent (µ σ,j = µ j , ∆µ = µ L − µ R = 0), there is no spin current and the noise is related to the mass current by S(0) = coth[∆µ/(2T )]I mass .