Long range mediated interactions in a mixed dimensional system
Daniel Suchet Zhigang Wu Frédéric Chevy Georg M. Bruun

  
  




  
   
        
   










    ρχr̊ a b c w d k p q r v x z G R  H F J P⟨⟩ℋℳ𝒢ℒ𝒱ℰ𝒵ψ̂ψ̂^† Trϵε∂τk̅r̅n̅Laboratoire  Kastler  Brossel,  ENS-PSL  Research  University,CNRS,  UPMC,  Collège  de  France,  24,  rue  Lhomond,  75005  Pariszwu@mail.tsinghua.edu.cnInstitute for Advanced Study, Tsinghua University, Beijing, 100084, ChinaLaboratoire  Kastler  Brossel,  ENS-PSL  Research  University,CNRS,  UPMC,  Collège  de  France,  24,  rue  Lhomond,  75005  ParisDepartment of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
    We present a mixed-dimensional atomic gas system to unambiguously detect and systematically probe mediated interactions. In our scheme, fermionic atoms are confined in two parallel planes and interact via exchange of elementary excitations in a three-dimensional background  gas. This interaction gives rise to a frequency shift of the out-of-phase dipole oscillations of the two clouds, which we calculate using a strong coupling theory taking the two-body  mixed-dimensional scattering into account exactly. The shift is shown to be easily measurable for strong interactions and can be used as a probe for mediated interactions.
    
     Long range mediated interactions in a mixed dimensional system
        Georg M. Bruun
        December 30, 2023
    ==============================================================
    
    
    Mediated interactions  were originally introduced to provide a quantum-mechanical explanation for the peculiar “action at a distance"  interactions like gravity and electromagnetism and they now constitute a major overarching paradigm in physics. In particle physics, exchange of gauge bosons is responsible for the propagation of fundamental interactions <cit.>. In condensed matter, the attraction between the electrons in BCS superconductors arises from the exchange of lattice phonons <cit.>, and it is speculated that the mechanism
    behind high-T_c superconductivity lies in the exchange of spin fluctuations <cit.>. The concept of mediated interactions is also important in classical physics, where fluctuations of classical fields are responsible for phenomena such as the finite-temperature Casimir effect in electrodynamics <cit.> and in  biophysics <cit.>.
    
    
    Ultracold atoms have emerged as a versatile platform for the  investigation of many-body physics, and a host of schemes have been proposed to explore mediated interactions using these systems. For instance, mediated interactions lead to the formation of a p-wave superfluid  in spin-imbalanced fermionic systems <cit.>; they are responsible for the formation of a topological superfluid with a high critical temperature in 2D systems <cit.>, and in 1D quantum liquids they are shown to result in Casimir-like forces between impurities <cit.>. In most cases, however, the mediated interaction is weak
    and in competition with direct interactions between atoms, making its experimental observation challenging.
    
    
    
    In this paper, we apply the mixed-dimensional setup proposed in <cit.> and illustrated in Fig. <ref> to study mediated interactions.  Specifically we consider two parallel layers located at z_1=0 and z_2=d, which contain an equal number of spin-polarized non-interacting fermions (A-species). The layers are immersed in a uniform 3D gas of interacting spin 1/2 fermions (B species), which can be tuned through the BEC-BCS cross-over. The presence of the 3D gas induces a mediated interaction between the A-particles: one A-particle will perturb locally the surrounding B-particles thereby inducing excitations in the 3D gas, which in turn affects the dynamics of a second A-particle. If  A-particles are harmonically trapped, this mediated coupling leads to a beating between oscillations in the two planes. Measuring the beating frequency between the 2D-clouds therefore gives access to the strength of mediated interaction. This scheme is similar to Coulomb drag experiments in bilayered electronic systems <cit.> that was recently generalized to the case of dipolar gases <cit.>.
    
    To analyze the dynamics of this system, we develop a systematic many-body theory for the  mediated inter-plane interaction that includes the low-energy mixed-dimensional A-B scattering exactly.  We then derive an  expression for the associated interaction energy between the two planes and calculate
     the frequency of the out-of-phase dipole oscillations of the 2D clouds in the xy-plane. In the weak A-B interaction limit, our results recover the perturbative expression for a mediated interaction proportional to the  density-density response function of the 3D gas.
     In the strong A-B interaction limit, however, the weak-coupling result breaks down completely. In the latter case we focus on the BEC regime of the 3D gas and show that the mediated interaction gives rise to a significant and easily detectable shift in the out-of-phase dipole oscillation frequency of the two clouds.
    
    2D-3D scattering.–
    The interaction between the A and B particles is short range and
    can  be characterised by an effective 2D-3D scattering length a_ eff <cit.>. Solving for
    the scattering matrix in the many-body medium yields
    
        𝒯_AB(_⊥,iω_ν) = g/1- gΠ(_⊥,iω_ν),
    
    where  g=2π a_eff/√(m_Bm_r), and m_r=m_Am_B/(m_A+m_B) is the reduced mass (ħ=k_B=1). Here m_A denotes the mass of an A-fermion and m_B that of the scattering particle in the 3D gas, namely the mass of B-fermion (dimer) in the BCS (BEC) regime. Π(_⊥,iω_ν) is the renormalised 2D-3D pair propagator for the center-of-mass (COM) momentum _⊥=(p_x,p_y) in the plane,  and  iω_ν is either a bosonic (BCS regime) or fermionic (BEC regime) Matsubara frequency.  Equation (<ref>) includes many-body effects in the ladder approximation (see the Supplemental Material), and recovers the correct low energy  2D-3D scattering matrix in a vacuum <cit.>. .
    
    
    Mediated interaction for weak 2D-3D interaction.–
    Consider first the case of a weak 2D-3D interaction where a_eff is much smaller than the interparticle spacing of the A and B particles.
    We  then have T_AB(_⊥,iω_ν)≃ g from  (<ref>), and second-order perturbation theory gives
    
        V_ m.i.(𝐪_⊥,iω_ν)=g^2∫_-∞^∞d q_z e^iq_zdχ_B(_⊥,q_z,iω_ν),
    
    which describes the mediated interaction between two A-particles in  different planes. Here (_⊥,iω_ν)=(q_x,q_y,iω_ν)
     are the transferred momentum and frequency and  χ_B(_⊥,q_z,iω_ν) is the density-density response function of the  B-cloud.
     The integration over the momentum q_z  comes from the fact that it is not conserved in the 2D-3D scattering. Deep in the BCS  limit where the
    B fermions form an ideal Fermi gas,  the mediated interaction (<ref>) is of the form of a Ruderman-Kittel-Kasuya-Yosida potential <cit.>.
    When the B fermions are deep in the BEC limit where they form a weakly interacting BEC of dimers, the mediated interaction takes the form of a Yukawa  potential <cit.>. At zero frequency, Fourier transforming (<ref>) back to the real space gives
    
    V_ m.i.(r) =
    
      g^2m_B /16π^32p_Frcos2p_Fr-sin2p_Fr/r^4   BCS limit
    
      -g^2n_Bm_B /π re^-√(2)r/ξ_B   BEC limit
     where p_F is the Fermi momentum of the 3D Fermi gas in the BCS regime, and  n_B is the density of the 3D BEC of dimers with coherence length ξ_B=1/√(8π n_B  a_B).
     Here, a_B=0.6a_BB is the  scattering length between the  deeply bound dimers of B fermions <cit.>.
    
    Mediated interaction for strong 2D-3D interaction.–
    For a strong 2D-3D interaction where a_eff is comparable to or larger than the interparticle spacing, the mediated interaction between the two layers  takes on a more complex form. The reason is that we need to retain the  full COM momentum and frequency dependence of the 2D-3D scattering matrix given by (<ref>).
    
     We shall from now on concentrate on the BEC limit of the B-fermions, namely when they form a weakly interacting BEC of dimers, which can be treated within Bogoliubov theory.
     The mediated interaction between the A-particles is calculated including
     all processes where a single Bogoliubov phonon in the BEC is exchanged between  the two layers.
      In a diagrammatic language, these processes are  shown in Fig. <ref> (a).
    
    Summing up the contributions from the four terms  in Fig. <ref> (a) gives
    
        V_ m.i.(p_1,p_2;q)     = n_B𝒯_AB(p_1+q)𝒯_AB(p_2) G̅^B_11(𝐪_⊥,iω_ν)
        
        +   n_B𝒯_AB(p_1)𝒯_AB(p_2-q) G̅^B_11(-𝐪_⊥,-iω_ν)
        
        +   n_B𝒯_AB(p_1+q)𝒯_AB(p_2-q) G̅^B_12(𝐪_⊥,iω_ν)
        
        +   n_B𝒯_AB(p_1)𝒯_AB(p_2) G̅^B_21(𝐪_⊥,iω_ν),
    
    where  p_1≡ (_1⊥,iω_m_1), p_2≡ (_2⊥,iω_m_2), and q ≡ (_⊥,iω_ν).
    Here ω_m = (2m+1)π /β and ω_ν = 2νπ /β are Fermi and Bose Matsubara frequencies respectively, where β=1/T is the inverse temperature and  m and ν are integers.
    In  (<ref>), the Green's functions of the BEC are integrated over the z-component of the momentum as
    
        G̅^B_αβ(𝐪_⊥,iω_ν)≡∫_-∞^∞dq_z/2πG^B_αβ(𝐪_⊥,q_z,iω_ν)e^iq_z d.
    
    The Green's functions of the 3D BEC are as usual
    
    G^B_11 = u_^2/iω_ν-E_-v_^2/iω_ν+E_, G^B_12=g_Bn_B/ω_ν^2+E_^2
    where = (_⊥,k_z) and G^B_21(,iω_ν)=G^B_12(,iω_ν). We have defined
       u_^2, v_^2 =1/2[  (_ + g_Bn_B)/E_± 1 ], E_ = √(_ (_ + 2 g_Bn_B) ) is the Bogoliubov spectrum with
      _ = k^2/2m_B, and  g_B = 4π a_B/m_B. Note that the mediated interaction
       (<ref>) depends on both  p_1 and p_2 as well as q due to the momentum and frequency dependence of the
     2D-3D scattering. In fact, in the weak interaction limit 𝒯_AB≃ g, one recovers (<ref>) from the more general expression (<ref>). 
    
    
    Thermodynamical potential.–
    We now derive an expression for the correction to the thermodynamic potential Ω due to the mediated interaction
    between the two planes for a general strength of the  2D-3D interaction. The dominant contribution is the Hartree term illustrated in Fig. <ref> (b). For a
    homogeneous system, this term gives the correction per unit area as(for the rest of the paper the ⊥ subscript will be dropped in the vector notation and all bold face letters now denote in-plane 2D vectors)
    
        Ω̅_ m.i. = 1/β^2∑_m_1 m_2∫d^2p_1/(2π)^2d^2p_2/(2π)^2V_ m.i.(p_1,p_2;0) 
        × G_1^A(_1,iω_m_1)G^A_2(_2,iω_m_2),
    
    where 
    G^A_j(,iω_m) = 1/(iω_m-p^2/2m_A +μ_A)
    is the Green's function for the A-fermions in the j-th layer with  μ_A being the chemical potential. Using (<ref>)
    together with the identity 2G̅_11(0,0)+2G̅_12(0,0)=-√(2)n_Bm_Bξ_Bexp(-√(2)d/ξ_B) yields
    
        Ω̅_ m.i. =-√(2)m_Bξ_B n_B e^-√(2)d/ξ_BΩ̅_1 Ω̅_2,
    
    where
    
        Ω̅_j = 1/β∑_m∫d^2p/(2π)^2𝒯_AB(,iω_m)G^A_j(,iω_m).
    
    We point out that the Matsubara frequency summation in the above expression can in fact be performed analytically (see supplementary material), which greatly simplifies the numerical calculation of thermodynamic potential density. 
    
    Local-density approximation.– Using the local-density approximation, we can
     generalize (<ref>),
     which was derived assuming homogeneous system, to the case of trapped 2D Fermi clouds. This yields the total correction as 
      
        Ω_ m.i.(_1-_2) =∫ d^2r_1d^2r_2[2G̅^B_11(_1-_2,0) 
        
        +2G̅^B_12(_1-_2,0)] Ω̅_1 (_1-_1)  Ω̅_2 (_2-_2),
    
    where  G̅^B_ij(,0) is the Fourier transform of G̅^B_ij(,0) back to real 2D space,
    and Ω̅_i () is given by (<ref>) using a local chemical potential μ_A()=μ_A+m_Aω_⊥^2r^2/2.
    In (<ref>), we have allowed the two A-clouds to be rigidly displaced distances of _1 and _2 along the x-axis in order to analyse their coupled dipole
    oscillations, see Fig. <ref>.  Since G̅^B_ij already contains a Fourier transform with respect to z-momentum, see (<ref>),
    the bosonic Green's functions entering  (<ref>)  now simply add up  to the density-density correlation function of the BEC
    evaluated at the 3D real space distance r=|_1-_2+d|. Using this, we finally obtain
    
        Ω_ m.i.(_1-_2)=   -m_Bn_B/π∫ d^2r_1d^2r_2e^-√(2)r/ξ_B/r
           ×Ω̅_1 (_1-_1)  Ω̅_2 (_2-_2).
    
    Equation (<ref>) can be understood as follows. Consider two area elements of the 2D gases,  one located at _1-_1 in  layer 1 and the other at _2-_2 in layer 2. The contribution from these two elements can be approximated by the expression in (<ref>) in which the relative distance is taken to be r instead of d. Equation (<ref>) then sums up all such contributions in the two clouds.
    
    For weak interaction, we see from (<ref>) that Ω̅_j(_j-_j)=gn_j(_j-_j), where n_j (_j-_j) denotes the equilibrium fermion density in layer j rigidly displaced the distance _j along the x-axis. Equation  (<ref>) then simplifies to
    
        Ω_ m.i.(_1-_2)=   -g^2m_Bn_B/π∫ d^2r_1d^2r_2e^-√(2)r/ξ_B/r
           × n_1 (_1-_1) n_2(_2-_2),
    
     which  is the usual Hartree approximation for the interaction energy between the two planes mediated by a Yukawa interaction.
    
    
    Coupled dipole oscillations.–
    Consider now the situation where the two clouds perform dipole oscillations around their equilibrium positions, see Fig. <ref>.
    For small displacements _1 and _2, the COM velocities and the beating frequencies are  small compared to the speed of sound in the 3D gas and the trapping frequencies respectively, yielding rigid and undamped oscillations of the 2D clouds <cit.>. The COM dynamics is then determined by the energy increase δ E  associated with the  displacements of the clouds.
    For rigid displacements, we have
    δ E = Ω_ m.i.(_1-_2)-Ω_ m.i.(0)+ [μ_A(_1)+μ_A(_2)-2μ_A]N_A, which gives
    
        δ E(_1,_2)=  1/2N_Am_Aω_⊥^2(_1^2 +  _2^2)  + Ω_ m.i.(_1-_2)-Ω_ m.i.(0),
    
    where  N_A is the number of fermions in each layer.
    Taylor expanding Ω_ m.i.(_1-_2) to second order in _1-_2, we readily see that the motion of the two clouds separates into an in-phase oscillation with
    frequency ω_⊥, and an out-of-phase oscillation with frequency
    
        ω_r = ω_⊥√(1+ 2I/N_Am_Aω_⊥^2),
    
    where
    
    I = .^2/_1^2Ω_ m.i.(_1-_2) |__1-_2=0.
    
    The microscopic expression for ω_r for arbitrary strength of the  2D-3D interaction in terms of (<ref>), (<ref>), (<ref>), and (<ref>) is the  main result of this letter and it explicitly shows how the mediated interaction can be probed by measuring the frequency of the out-of-phase
    dipole oscillations of the two clouds.  
    
    
    
    Results.–
    We now calculate the frequency  ω_r for a realistic cold-atom system consisting of  N_A = 1000^40 K atoms  trapped in each plane, immersed in a 3D BEC of ^6 Li dimers.
      The  transverse trapping frequency
     for the  ^40 K clouds is ω_⊥ = 2π× 380   Hz, the density of the BEC is n_B=10^18  m^-3, and the coherence length is  ξ_B= 2.7  μ m. We furthermore assume that the temperature is zero. In Fig. <ref>, we show the frequency  ω_r/ω_⊥ as a function of the 2D-3D interaction strength  1/k_Fa_ eff at a fixed interlayer distance d = 0.4  μ m.
      The frequency increases monotonically as a_ eff increases. For weak interaction, it agrees with the second order result (dashed line). For stronger interaction, the full frequency/momentum dependence of the 2D-3D scattering is important, and the perturbative result deviates significantly from the full strong-coupling theory. In particular, whereas the perturbative result
     diverges for 1/k_Fa_ eff→ 0, the strong-coupling theory predicts a finite frequency saturating at ω_r≃ 1.48  ω_⊥. Importantly, the frequency  shift
      becomes significant for -2≲1/k_Fa_ eff≤ 0, which includes a region sufficiently far from unitarity so that
       the predicted 3-body loss is small <cit.>. This demonstrates   the usefulness of our proposal to detect  mediated interactions.
       Note that this result can only be obtained using a strong coupling theory, since the perturbative result is only accurate for weak interactions where the frequency shift is minute.
    
    
    In Fig. <ref>, we plot ω_r/ω_⊥ as a function of the ratio of the interparticle distances n_B^1/3/n_F^1/2 (keeping n_F fixed)  with 1/k_Fa_ eff=-0.1 and all other physical  parameters the same as for  Fig. <ref>.a. The density of the BEC enters the mediated interaction in two ways, which is most clearly seen in the weak-coupling limit given by  (<ref>): First, the strength of the  interaction is  proportional to n_B;  second, the range of the  interaction is determined by the BEC coherence length ξ_B∝ 1/√(n_B). Thus, increasing the density increases the strength but reduces the range of the mediated interaction, and it is not a priori obvious what the net effect on the frequency shift will be. From Fig. <ref>, we see that for the chosen parameters, ω_r in fact increases monotonically with increasing BEC density [We restricted all figures to negative values of the 2D-3D scattering length. Indeed for 1/k_Fa_ eff> 0, a 2D fermion can form a bound-dimer state with a 3D boson.
    The frequency shift in this region therefore depends on whether the system forms these dimers, or whether it is  on the so-called repulsive branch where the effective 2D-3D interaction is repulsive. This complicates the analysis, which will be presented in a future publication.
    ].
    
    
    
     Conclusions.–
     We demonstrated that a mixed-dimensional setup consisting of two layers of identical fermions immersed in a 3D background gas is a powerful probe to investigate mediated interactions systematically. The mediated interaction between the two layers modifies the out-of-phase dipole oscillation frequency of the 2D clouds, and we calculate this shift using a strong-coupling theory taking into account the low energy scattering between the 2D and 3D particles. Using this theory, we showed that for strong 2D-3D coupling, the resulting frequency shift is clearly measurable.
    
    Finally we note that the advantages of our proposal are twofold. First, if the 2D trapping is realized using optical potentials, the distance between planes is a few hundred nanometres, which is much larger than the range of interatomic interactions. Any observed coupling between the two planes is therefore solely due to a mediated interaction via the 3D gas. Second, the shift of the center-of-mass oscillation frequency is a very precise spectroscopic tool that can be used as a probe of  weak interactions, as demonstrated recently in <cit.>.
    
    
    
    FC and DS acknowledge support from Région Ile de France (DIM IFRAF/NanoK), ANR (Grant SpiFBox) and European Union (ERC Grant ThermoDynaMix).
    GMB and ZW wishes to acknowledge the support of the Villum Foundation via Grant No. VKR023163.
    
    DS  and ZW contributed equally to this work.
    
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     Supplemental Material 
    
    Daniel Suchet^1, Zhigang Wu^2, Frédéric  Chevy^1, and G. M. Bruun^3
    
    ^1Laboratoire  Kastler  Brossel,  ENS-PSL  Research  University,CNRS,  UPMC,  Collège  de  France,  24,  rue  Lhomond,  75005  Paris 
    ^2Institute for Advanced Study, Tsinghua University, Beijing, 100084, China
    ^3Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
    
    
    
    § 2D-3D SCATTERING MATRIX
    
    We provide some details on the 2D-3D scattering matrix given in Eq. (1) in the main text.  In the strong 2D-3D interaction limit and in the presence of a 3D BEC background, we need the scattering amplitude in medium between the a 2D fermion and a 3D boson to determine the mediated interaction. In terms of the well-known T-matrix approximation, the scattering amplitude 𝒯_AB satisfies an integral equation represented diagrammatically in Fig. <ref>. Here the ⊥ subscript is used to distinguish 2D plane vectors from the 3D ones. Using standard procedure the scattering matrix can be expressed in terms of the 2D-3D zero-energy scattering amplitude in vacuum g=2π a_eff/√(m_Bm_r), where m_r = m_Am_B/M with M=m_A + m_B being the reduced mass and a_ eff being the 2D-3D scattering length. In doing so, it can be shown that 𝒯_AB only depends on the total momentum and frequency _⊥=_1⊥+_2⊥ = _3⊥+_4⊥ and ω_ν = ω_n_1 + ω_ν_2 =  ω_n_3 + ω_ν_4. We find
    
        𝒯_AB(_⊥,iω_ν) =  g/1- gΠ(_⊥,iω_ν).
    
    Here Π(_⊥,iω_ν) is the renormalised pair propagation given by
    
        Π(_⊥,iω_ν) = ∫d^3p'/(2π)^3[
        				u__+^21 + b(E__+) - f(ξ__-)/iω_ν - E__+ - ξ__-
        				+
        				v__+^2b(E__+) + f(ξ__-)/iω_ν + E__+ - ξ__-+1/p_z^' 2/2m_B+'^2_⊥/2m_r+i0^+ ],
    
    where '=('_⊥,p'_z), _+ ≡m_B/M_⊥+', _- ≡m_A/M_⊥ - '_⊥, and b(x) = 1/(e^β x-1) and f(x)=1/(e^β x+1) are the Bose and Fermi distribution function respectively.  For weakly interacting Bosons, it is a good approximation to replace the normal Green's function G^ B_11(,iω_ν) by the non-interacting Boson Green's function G^ B_0(,iω_ν) = 1/(iω_ν-_+μ_B) in the scattering T-matrix. With this simplification, we find at T=0
        Π(_⊥,iω_ν) = ∫d^3p'/(2π)^3[
        				1  - θ (k_F- |m_A_⊥/M-'_⊥ | )/iω_ν -  ( _⊥^2/2M+_⊥^' 2/2m_r+p_z^' 2/2m_B )+μ_A+1/p_z^' 2/2m_B+^' 2_⊥/2m_r+i0^+ ],
    
    where k_F = √(2m_Aμ_A) is the Fermi momentum of the A-species. 
     
    Expressed in terms of the  dimensionless variables, the pair propagator is 
    
        Π(_⊥,iω_ν) = 2m_A k_F∫d^2 p_⊥'/(2π)^2∫d p'_z/2π[
        				1  - θ (1- |'_⊥-α_A_⊥ | )/iω_ν -  ( α_Ap_⊥^2+α_B^-1p_⊥^' 2+α_Aα_B^-1p_z^' 2 )+1+1/α_Aα_B^-1p_z^' 2+α_B^-1p_⊥^' 2+i0^+ ],
    
    where  α_A = m_A/M and α_B = m_B/M.  Here the frequency variables are scaled in terms of the chemical potential μ_A and the momentum variables in terms of the Fermi momentum k_F. We write 
    Π(_⊥,iω_ν)  =  Π_0(_⊥,iω_ν) + ΔΠ(_⊥,iω_ν),
    
    where
    
        Π_0(_⊥,iω_ν)     ≡ 2m_A k_F∫d^2 p'_⊥/(2π)^2∫d p'_z/2π[
        				1 /iω_ν -  ( α_Ap_⊥^2+α_B^-1p_⊥^' 2+α_Aα_B^-1p_z^' 2 )+1+1/α_Aα_B^-1p_z^' 2+α_B^-1p_⊥^' 2+i0^+ ] 
            = -im_A k_F/2πα_A^1/2α_B^-3/2√(iω_ν+1-α_Ap_⊥^2)
    
    is the pair propagator in vacuum and 
    
        ΔΠ(_⊥,iω_ν)    ≡ -2m_A k_F∫d^2 p'_⊥/(2π)^2∫d p'_z/2πθ (1- |'_⊥-α_A_⊥ | )/iω_ν -  ( α_Ap_⊥^2+α_B^-1p_⊥^' 2+α_Aα_B^-1p_z^' 2 )+1
            = im_A k_F/√(α_Aα_B^-1)∫d^2 p'_⊥/(2π)^2θ (1- |'_⊥-α_A_⊥ | )/√(iω_ν+1-α_Ap_⊥^2-α_B^-1p_⊥^' 2)
    
    is the medium correction. Here √(z) always denotes the root of the complex number z that lies in the upper half plane. 
    
     From Eq. (<ref>)-(<ref>) we find (from now on we drop the ⊥ sign from the 2D vectors)
    Π(,iω_ν) = -im_A k_F/2π^2α_A^1/2α_B^-3/2∫_0^π/2 dθ (√(iω_ν-γ_+(θ,p)) -√(iω_ν-γ_-(θ,p) ) ) 
    
    for α_A p ≤ 1. Here and in the following
    γ_± (θ,p) ≡α_B^-1p^2_±(θ)+α_Ap^2-1,
      
    where
    p_±(θ) = ±α_A p cosθ + √(1-α^2_A p^2 sin^2 θ). 
    For α_A p > 1 we find
    
        Π(,iω_ν) =    -im_A k_F/2π^2α_A^1/2α_B^-3/2[ √(iω_ν-(α_Ap^2-1)) +1/π∫_0^θ_0 dθ (√(iω_ν-γ_+(θ,p)) -√(iω_ν-γ_-(θ,p) ) )  ],
    
    where θ_0 = sin^-1 (1/α_A p).
    
    
    
    § CALCULATION OF  Ω̅_J
    
    We now determine Ω̅_j given in Eq. (9) in the main text, which is reproduced below 
    
        Ω̅_j = 1/β∑_m∫d^2p/(2π)^2𝒯_AB(,iω_m)G^A_j(,iω_m).
    
    In terms of the dimensionless momenta and frequencies introduced earlier, we get 
    
        Ω̅_j    =  2gm_A/β∑_m∫d^2p/(2π)^21/ [1-gΠ(,iω_m) ] [ iω_m-(p^2-1)]
            =  gm_A/πβ∑_m∫_0^∞ dp   1/ [1-gΠ(,iω_m) ] [ iω_m-(p^2-1)]
    
    Submitting Eq. (<ref>) and (<ref>) into Eq. (<ref>), and performing the Matsubara frequency summation, we find for negative 2D-3D scattering length a_ eff < 0 and in the zero temperature limit β→∞
        Ω̅_j     = 2 a_ effα_A^1/2α_B^-1μ_A∫_0^1 dp S(p),
    
    where 
    
        S(p)    = 1/1-k_Fa_ eff√(α_B)1/π∫_0^π/2 dθ [√(1-p^2 +γ_+(θ,p)) +√(1-p^2+γ_-(θ,p)) ].
