Spin-selective scatterers as a probe of pairing in a one-dimensional interacting fermion gas

We study two species of attractively interacting fermion confined to a quasi-one-dimensional geometry, in the presence of a strong scattering potential that can couple, selectively, to one or both species. We show that the fermion density distribution in the presence of such a spin-selective scattering potential reflects the pairing spin gap of the fermions.


Introduction
In recent years, there has been much excitement surrounding the achievement of correlated phases of matter (like those found in condensed-matter systems) in cold-atom experiments [1,2]. Part of the excitement stems from the extreme controllability of such coldatom experiments, with parameters such as the interactions, particle densities and even spatial geometry being experimentally adjustable. This controllability provides a large parameter space of interacting systems to study, enhancing the comparison between theory and experiment.
In this paper we study the properties of two species of fermion confined to a quasi-onedimensional (1D) geometry (achievable via a highly prolate trapping potential), with attractive 2 short-ranged interactions that lead to pairing correlations. Recent experiments have studied attractive fermions in the 1D limit [3], in the case of an imposed population imbalance that can have an imbalanced superfluid phase that is predicted [4]- [9] to be a 1D analogue of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [10,11], in which the population imbalance induces a spatial modulation in the local pairing correlations. One open question concerns how the FFLO pairing correlations in imbalanced 1D gases would be observed experimentally.
Here, we focus on a balanced gas, but consider the case of an imposed potential that couples differently to the two fermion species, and which can create a local population imbalance. The motivation for studying this system follows from the fact that, in recent years, experimentalists have begun to explore spin-dependent trapping potentials [12,13], and it is natural to consider how generic spin-dependent potentials can probe interacting Fermi gases. Recent theoretical work has investigated the properties of fermions in spin-dependent optical lattices [14]- [18] and in the case of separate harmonic trapping potentials for the two spin states [19], predicted to locally induce FFLO-like oscillatory pairing correlations. In this paper, we consider the case of a spatially inhomogeneous trapping potential that forms a local 'bump' or impurity at the spatial origin. Such systems have been studied for many years [20]- [24], mainly in the context of a local impurity or barrier in a 1D electron gas. We also note recent theoretical work studying the effect of impurity potentials on 1D cold atomic gases [25,26]. In this paper, we study a spin-selective local scattering center that one can imagine suppresses local singlet pairing in the 1D gas, thereby probing such pairing correlations.

Model Hamiltonian
Our model Hamiltonian involves two species of interacting fermion (ψ σ ) confined to a quasi-1D limit: where λ < 0 measures the strength of the short-range attractive interspecies interactions, ρ σ = |ψ σ | 2 , k F is the Fermi wavevector, and V σ (x) represents a short-ranged, possibly spindependent, scattering potential centered at the origin x = 0. In a cold-atom experiment involving a highly prolate trapping geometry, λ can be related to an underlying effective 1D scattering length a 1D via which, in turn, can be related to the (experimentally controllable) 3D scattering length and the trap oscillator length [27]. We will analyze this system using the bosonization approach, which has the advantage of being technically straightforward while still retaining nontrivial interaction effects. We thus express the fermions in terms of bosonic fields θ σ and φ σ as [23,24,28,29] ψ where ρ 0 is the mean fermion density, related to the Fermi wavevector via k F = πρ 0 /2. In the first line, the subscripts R and L refer to right-and left-moving fermions. As these expressions 3 involve expansions for momenta near the Fermi surface, they only apply at sufficiently low energies, below a scale of the order of the Fermi energy. The bosonized form of the fermion densities are given by It is standard to introduce charge (φ c ) and spin (φ s ) versions of φ ↑,↓ , defined by Using analogous expressions to define θ c and θ s , and switching to an effective action for the bosonic fields, we find where S imp is the impurity action and the Luttinger interaction parameters and velocities take the well-known form with the + (−) being associated with c (s) in each line, so that K s < 1, K c > 1, and u s > u c for the present case of attractive interactions (λ < 0), reflecting spin-charge separation. Here, g = 1 2 λρ 2 0 and v F =hk F /m is the Fermi velocity. In the bulk, the attractive interactions lead to a 1D paired superfluid state, pinning φ s ≈ 0 (representing the spin gap). Our aim is to see how the impurity, localized at x = 0, alters the local fermion densities and pairing correlations. We consider two kinds of impurity: symmetric (which couples to both species equally) and asymmetric (which couples to to only one species, that we will always take to be the ↓ species). While an symmetric impurity can be realized by locally modifying the prolate trapping potential (e.g. a local constriction), an asymmetric impurity would require a spin-dependent trapping potential. We also restrict attention to the strong-scattering limit, in which the impurity potential V σ effectively pushes ρ σ → 0. In the symmetric impurity case, this constrains φ σ (0) = π/2 at the impurity center (x = 0) for both σ =↑ and σ =↓, whereas for the asymmetric case we have φ ↓ (0) = π/2 with φ ↑ unconstrained.

Symmetric scattering potential
We begin with the case of a strong symmetric scattering potential that couples equally to the spins-↑ and the spins-↓. In the language of the spin and charge fields, this constrains φ c (0) = α c = π/ √ 2 and φ s (0) = α s = 0. In the bulk, the field φ s (x, τ ) is expected to exhibit only small fluctuations around the minimum value arising from the cosine term in equation (8). Given this, we can approximate this term by an effective gap 2 representing the energy penalty towards pair breaking, i.e. the underlying spin gap. An estimate for is provided below in section 3.
Our aim is to calculate the local density and magnetization near the scattering center, given by (neglecting the derivative term) with the angle brackets representing the average with respect to the action S. Our approach, following [21,22], involves implementing the above constraints with Lagrange multiplier fields λ i (τ ) (with i = c, s) that pin φ c and φ s to the values α c and α s consistent with the impurity potential. Integrating out the θ i fields, we have the effective action S = S c + S s + S λ with It is sufficient to consider the averages e iβφ i (x) with i = c, s. These decouple since, in the present case, the symmetric scatterer preserves spin-charge separation. Beginning with the φ c average, we have with S λ c being the part of S λ containing λ c and the demoninator Z being the other factor with β = 0. Evaluating the functional integral over φ c introduces the corresponding Green function (note we always take temperature T → 0) We then obtain with the effective action for the Lagrange multiplier field Here, we have used the fact that G c (x, ω) is even in its arguments. The remaining functional integral can be evaluated by considering the equation of motion for λ c (ω): which, when inserted back into equation (19) and using equation (18), yields where the first term in square brackets should be interpreted as lim ω→0 G c (x, ω)/G c (0, ω) = 1. The second integral is formally divergent, but should be regularized [21,22] at the ultraviolet (UV) scale by introducing the cutoff (by multiplying the integrand by exp[−ω/ ]). This gives which has the correct limiting behavior for x → 0 (consistent with φ c (0) = α c ). The second average we need is for the spin field, e iβφ s (x) , subject to the constraint φ s (0) = α s . However, examining equation (14) it is clear that the only difference relative to the preceding calculation is the fact that the φ s action now has a mass due to the presence of pairing. Thus, the result must be the same but with the Green function G c replaced by the corresponding spin Green function sensitive to the pairing scale . With this, we have the effective action for λ s : and, using lim ω→0 G s (x, ω)/G s (0, ω) = exp[−|x| /u s ], we find for the necessary average: which again is consistent with the constraint φ s (0) = α s . We now proceed to use these averages, along with equations (11) and (12), to determine the density response in the presence of such a strong symmetric scatterer. For the correlator involving the spin field, we clearly need equation (26) with α s = 0 and β = ± √ 2; however, since e i √ 2φ s (x) = e −i √ 2φ s (x) , we immediately have M(x) = 0 everywhere, as expected, since the scatterer couples equally to the two fermion species. Turning to the correlator involving the charge field, examining equation (22) with α c = π/ √ 2 and β = ± √ 2, and using equation (26), we find for the density where I (x) is proportional to the integral in equation (26), and given by Although I (x) cannot be easily evaluated analytically, it basic behavior is straightforward. For x → 0, I (x) vanishes (consistent with φ s (0) = 0), while, for x → ∞, I (x) ln / (reflecting the small Gaussian fluctuations of φ s around zero), so that we finally have valid for x → ∞, describing the asymptotic Friedel oscillations in this 1D paired-fermion gas [21,22,30] The power law of the envelope reflects the superfluid stiffness Luttinger parameter K c , implying that the long-distance response to such a symmetric scatterer is due to low-energy phase fluctuations of the superfluid field θ c . The result equation (29) agrees with well-established literature predicting that a strong impurity should lead to Friedel oscillations in the local density [21,22,30,31] of a 1D gas. We note that the precise exponent −K c /2 follows from the results of [30] upon taking the spin Luttinger parameter (K s in our notation) to vanish to account for the spin gap.

Magnetic scattering potential
In the preceding section, we found that a strong scattering potential that couples equally to the two fermion species leads to a suppression of ρ(x) that oscillates with wavevector 2k F , along with an overall envelope that decays away from the origin as a power law ∼ |x| −K c /2 . In contrast, in the present section, we show that an asymmetric or 'magnetic' scattering potential, that couples preferentially to one spin species, shows quite a different spatial profile that is sensitive to the underlying spin gap of the system. Once again, we consider the case of a strong scattering potential, but assume it couples only to the spins-↓, effectively constraining φ ↓ (0) = π/2 (so that, near x → 0, ρ ↓ → 0). Then we have (once again integrating out θ c and θ s ) the effective action with the parameter α = π/2. To obtain the spin-↑ and spin-↓ densities, we need to compute the average The difference relative to the preceding section is clear. Previously, we had separate Lagrange multiplier fields for charge (λ c ) and spin (λ s ), and we separately integrated out the φ c and φ s fields to obtain effective actions for λ c and λ s . In the present case, we have only one Lagrange multiplier field λ that couples to both φ c and φ s in a way similar to the preceding section but with the rescalings λ c → λ/ √ 2 and λ s → −λ/ √ 2. This implies, upon integrating out the φ c and φ s fields, an effective action for λ (by analogy with equations (19) and (25)) Following the procedure of the preceding section, the equation of motion for λ(ω) leads to which we must insert into equation (32) to get the final result for equation (31). At this point, we focus on the cases of β s = ±β c , which amounts to considering the corresponding averages e i √ 2βφ ↑,↓ for φ ↑,↓ . We obtain with I σ (x) given by where ↑= + and ↓= − on the right side. Equation (35) satisfies I ↓ (0) = 0, consistent with the constraint φ ↓ (0) = α when inserted in equation (34). Using this result for I σ along with equations (11) and (12), we find where δρ(x) and δ M(x) represent the envelope functions of the local density and magnetization, given by δ M(x) = −e 1/4(I ↑ (x)+I ↓ (x)) sinh 1 4 (I ↑ (x) − I ↓ (x)) .
Although the integrals I ↑ (x) and I ↓ (x) must be calculated numerically, it is straightforward to determine the asymptotic large x behavior of these functions, and thus of ρ(x) and M(x), showing how the spin-gap energy scale would emerge in an experiment. We find that the sum I ↑ (x) + I ↓ (x) 2K c ln u c |x| , whereas the difference satisfies I ↑ (x) − I ↓ (x) ∝ exp(−|x|/L )/(|x|/L ), decaying exponentially over a length scale L ≈ v F / that directly reflects the underlying spin gap in this 1D Fermi gas (in this expression, we have assumed u s u c v F for simplicity). These limiting results, plugged into equations (38) and (39), imply a rapid vanishing of δ M(x) beyond the length scale L , but a slow power-law variation of δρ(x) with increasing x UV scale to be equal to the Fermi energy f =h 2 k 2 F /2m, the spin gap 0.3 f , and the coupling parameter entering equations (9) and (10) to be λ/πv F −0.5.
We can try to estimate the magnitude of in present-day experiments by using the standard Bardeen-Cooper-Schrieffer mean-field theory, which gives ≈ f exp . Plugging in equation (2) gives the argument of the exponential function to be −π k F a 1D /2. Using the reported value of a 1D ≈ 2099a 0 in [3] (with a 0 the Bohr radius), and estimating k F = 5.5 × 10 6 m −1 (from the reported Fermi energy), we find this exponent to be ≈ −0.96, implying / f 0.38, a strong pairing gap with the associated length scale L 0.95 µm To conclude, we have shown that, in response to a strong spin-selective scattering potential, the total density responds analogously to the case of a symmetric impurity (i.e. with the same power law as in equation (29)), while the local magnetization or population imbalance returns to its equilibrium value over a length scale governed by the pairing spin gap parameter . There are several remaining questions for future research, including the behavior away from the asymptotic strong scattering limit, as well as the generalization to the case of an imbalanced gas. Regarding the latter problem, one may imagine that such a spin-selective scattering potential may pin or enhance the oscillatory pairing correlations in the 1D FFLO phase (that have been detected in computational theory [7,8] but not yet in experiments), making them easier to observe. Finally, there is the issue of the harmonic trapping potential that is always present in experiments but ignored here. We expect that, as long as the spatial variation of the atom density due to the trap (i.e. the Thomas-Fermi length) is large compared to the density variations induced by the scatterer, our principal conclusions will remain intact. However, an important direction for future research would be to study this claim in detail and examine how the trapping potential impacts the Friedel oscillations induced by local scattering potentials.