An effective many-body theory for strongly interacting polar molecules

We derive a general effective many-body theory for bosonic polar molecules in strong interaction regime, which cannot be correctly described by previous theories within the first Born approximation. The effective Hamiltonian has additional interaction terms, which surprisingly reduces the anisotropic features of dipolar interaction near the shape resonance regime. In the 2D system with dipole moment perpendicular to the plane, we find that the phonon dispersion scales as $\sqrt{|\bfp|}$ in the low momentum ($\bfp$) limit, showing the same low energy properties as a 2D charged Bose gas with Coulomb ($1/r$) interactions.

in Sec. VI, and calculate the phonon mode dispersion as well as the Kosterlitz-Thouless transition temperature. We then summarize our work in Sec. VII.

II. LOW ENERGY SCATTERING THEORY OF DIPOLES
For the convenience of later discussions, we first briefly review the recent progress on the scattering problem of dipolar gas, where the electric/magnetic dipole moment is polarized by the external electric/magnetic field along z direction. The most general form of the scattering amplitude between two identical particles in such situation can be expressed to be where the scattering matrix element, t l ′ m ′ lm (k), depends on the relative incident momentum, k, and the summation is over even l for bosons and odd l for fermions. Y lm (r) is the spherical harmonic function of unit vectorr. At large distances, the inter-particle potential is dominated by the dipolar term, V d (r) = D 2 (1 − 3 cos 2 θ)/|r| 3 , where D is the electric dipole moment in c.g.s. unit (for simplicity, here we use electric dipoles to formulate the theory for polar molecules, while an similar version for magnetic dipolar gas can be also obtained easily); θ is the angle between the distance r and the dipole direction (polarized in z direction). However, at short distances the potential becomes much more complicated due to the Coulomb and spin exchange interaction between electrons. Deb and You [9] first calculated the cross section within a certain model potential and studied how they are changed near the shape resonance regime. Based on the numerical results, Yi and You [12] then proposed a pseudo-potential: to calculate the low energy scattering matrix element within the first Born approximation away from the shape resonance regime. Here a s = −t 00 00 (0) is the s-wave scattering length in zero field limit. Within the FBA [3,4], where a d ≡ M D 2 / 2 is a length scale and P 2 (x) is the Legendre Polynomial. θ k is the angle between the momentum k and z axis. As a result, the associated matrix elements become: ∞ krc dr r j l (r)j l ′ (r) with j l (r) being the spherical Bessel function. Here r c is a cut-off in the atomic length scale and therefore we can always take kr c → 0 in the low energy limit. In above FBA result, all the short-ranged effects are included in the s-wave part (a s ) only, while all other matrix element, t B l ′ m ′ lm (0), are proportional to the same length scale, a d ∝ D 2 . Therefore, it is easy to see why such results cannot be valid when the dipole moment (or external field) is sufficiently strong as higher order renormalization becomes important.
In Ref. [6], Derevianko developed a different pseudo-potential for dipolar interaction to go beyond the FBA. Although the most general expression of the pseudo-potential is derived for each scattering channels and the results are in principle applicable to strong interaction regime, but only one terms (the scattering between s-wave and d-wave, i.e. the t 20 00 (0) = t 00 20 (0) term in Eq. (1)) is evaluated within the leading order perturbative method (equivalent to the FBA level). In fact, we observe that Derevianko's result for the on-shell scattering channel (|k| = |k ′ |) is equivalent to the first two terms of the FBA result (i.e. the full f B (k, k ′ ) is replaced by −a s − a d 6 [P 2 (cos θ k ) + P 2 (cos θ k ′ )], using Y 20 (k) = 5/4πP 2 (cos θ k )). This explains why the meanfield calculation by Yi and You (Ref. [13], which included only the s-wave and s-d scattering channel of the pseudo-potential of Ref. [6]) is not consistent with the previous result even in the weak dipole moment regime, where the FBA is supposed to be valid. (We note that this inconsistence still exist even if Yi and You have ever used the corrected coefficient derived by the Erratum of Ref. [6]. The key point is that contributions from all other scattering channels are all proportional to D 2 within the FBA and hence cannot be neglected compared to t 20 00 (0).) In Ref. [8], Bortolotti et al. claimed that V ps (r) in Eq. (2) can be a good pseudo-potential if only one uses a dipole-dependent s-wave scattering length (i.e. a s (D)). However, their results cannot apply to the strong dipole moment regime when the shape resonance occur in other (different from s-wave) scattering channels due to the complicated electronic density distribution and/or spin exchange effect in a realistic polar molecule. Therefore, a general and useful approach to study the low energy many-body physics of strongly interacting polar molecules is still needed.

III. CRITERION FOR THE FIRST BORN APPROXIMATION
For completeness, now we explicitly examine the criterion for justifying the FBA in the low energy limit. We consider the following model potential: V mdl (r) = V d (r) for |r| > r c , and V mdl (r) = ∞ for |r| ≤ r c . Although (p,p',P, )= P 0 Γ  1: Series expansion for effective interaction in the ladder approximation. Solid line represents Green's function of bosonic particles and zig-zag line is for bare interaction. Here p ≡ 1 2 (p1 − p2) and p ′ ≡ 1 2 (p3 − p4) are the half the relative momentum, and P = p1 + p2 = p3 + p4 are the total momentum of the two scattering particles with frequency P0 (see also Ref. [16]). this model potential is over-simplified compared to the realistic interaction potential between polar molecules, it still catches the most important feature, anisotropic dipolar interaction, and hence should be useful in studying the validity of the Born approximation in the low energy limit. The full scattering wavefunction ψ(r), can be solved by: where ′ dr ′ is for |r ′ | > r c only, and is the exact scattered wavefunction for the hard core potential of radius r c without dipole moment. Here we have defined jn l (k, r) = j l (kr) − tan δ l (k)n l (kr) with δ l (k) ≡ tan −1 (j l (kr c )/n l (kr c )) being the scattering phase shift. j l (x) and n l (x) are the conventional spherical Bessel functions. The Green's function, G(r, r ′ ), satisfies ∇ 2 G(r, r ′ )+k 2 G(r, r ′ ) = −4πδ(r − r ′ ) with the boundary condition G(r cr , r ′ ) = 0, and therefore can be evaluated by using separation of variables. After some straightforward calculation, the Green's function can be expressed to be where r >(<) is the larger(smaller) one of r and r ′ , and h (1) l (x) ≡ j l (x) + in l (x). Within the FBA, the scattered wavefunction is given by the first order iteration: ψ B (r) = ψ 0 (r) − M 2 ′ dr ′ 4π G(r, r ′ )V mdl (r ′ )ψ 0 (r ′ ). Therefore its validity relies on the assumption that the change of the wavefunction is much smaller than ψ 0 (r) in the whole range of space [14]. We can therefore define a parameter, ξ, to measure the deviation of ψ B (r): ξ ≡ lim k→0 lim |r|→rc |ψ B (r) − ψ 0 (r)|/|ψ 0 (r)|. In such limit, we have ψ 0 (r) ∼ ∆r rc + O(kr c ), where ∆r = |r| − r c ≪ r c . Expanding the Green's function, G(r, r ′ ), in the small r regime, we obtain ψ B (r) − ψ 0 (r) = ∆r rc · π 3/2 3 √ 5 · MD 2 2 rc + O(kr c ). As a result, the condition to justify the FBA is ξ = π 3/2 3 √ 5 a d rc ≪ 1 [15]. For example, we consider the magnetic dipolar atom, 52 Cr, with r c ∼ 100a 0 as the typical length scale of van der Waals interaction. We find ξ Cr ∼ 0.4 < 1 and this explains why results obtained in the FBA for 52 Cr are comparable to experiments [3,4]. However, for polar molecules with electric dipole moment of the order of a few Debye, the value of ξ can easily be several hundred or more, where a shape resonance can occur in different channels to breakdown the FBA result. Therefore, in order to correctly describe the effective many-body physics of polar molecules, one needs a self-consistent theory beyond the pseudo-potential, V ps (r), and the first Born approximation.

IV. EFFECTIVE HAMILTONIAN IN 3D SPACE
To study the low energy physics of a general dipole interaction in the many-body medium, one has to use an effective two-particle interaction, Γ, which is just the vertex function integrating out all the contribution of virtual scattering in high energy limit [16]. A full calculation of the vertex function is usually not available (except in some special models of 1D systems), but can be well-approximated by using the standard ladder approximation (see Fig.  1). It is well-known that such ladder approximation is correct in the low density limit, and is therefore a very suitable approximation for systems of dilute cold atoms/molecules. Following the standard approach to evaluate the Bethe-Salpeter equation of bosonic particles [16], we can calculate the effective two-particle interaction (i.e. vertex function) within the ladder approximation by using the two-particle scattering amplitude, f (p, p ′ ): (6) where ǫ = M 2 P 0 − 2 P 2 /4M is the total kinetic energy in the center-of-mass frame; µ is the chemical potential and Ω is the system volume.
Using the fact, f (p ′ , k) * = f (k, p ′ ), the final term of Eq. (6) can be evaluated explicitly by integrating over the solid angle of momentum k in the scattering amplitude, Eq. (1). Furthermore, since the partial wave scattering matrix element, t l ′ m ′ lm (k), is known to be insensitive to the incident momentum, k, in the low energy limit, we can also neglect their momentum dependence and replace their value by a constant, t l ′ m ′ lm (0). As a result, the last term of Eq. (6) can be calculated to be where we have set m = m ′ due to the rotational symmetry about the polarization axis (z). It is easy to see that the real part of the integration cancels out, and the imaginary part proportional to 2M µ/ 2 in the limit of low energy scattering (|p ′ |, ǫ → 0). Therefore, the last term of Eq. (6) can be shown to be negligible when comparing with with the second term, f (p, p ′ ), in the low density limit, i.e. (|t l ′ m ′ lm (0)n Here n 3D is the 3D particle density). As a result, in the low energy and dilute limit, one can use Γ(p, p ′ ) = −4π 2 M f (p, p ′ ) as an effective "pseudo-potential" in momentum space (there is no dependence on total momentum and energy in such limit and we could omit them in Γ). Note that, different from the FBA used in the literature, we do not have to assume weak bare interaction in above derivation (strong interaction may still give small value of scattering matrix element, |t l ′ m ′ lm (0)|, in the low energy limit, just as in the usual s-wave scattering of cold atoms). Complicated electronic structure and shape resonance effects are all included in the full calculation (or experimental measurement) of the matrix elements, t l ′ m ′ lm (0) in all channels. In the rest of this paper, we will study the general effective theory and possible new many-body physics beyond the FBA without directly evaluating the scattering matrix elements.
Using the derived pseudo-potential (or effective interaction), Γ(p, p ′ ) = −4π 2 M f (p, p ′ ), we can write down the interacting Hamiltonian in momentum space by using the second quantization formalism: whereâ p andâ † p are field operator for bosonic polar molecules at momentum p. The momentum summation from now on is restricted to low momentum regime as implied by the effective interaction, Γ(p, p ′ ). In order to address the effect of pseudo-potential beyond the FBA (Eq. (2)), we can divide the contribution of pseudo-potential, Γ(p, p ′ ), into three parts: where the first term is from the known (dipole moment dependent) isotropic s-wave scattering, the second term is the usual FBA result for anisotropic dipolar interaction, the third term, f ∆ , is the scattering amplitude deviated from the known FBA results. It can be denoted to be with ∆a l ′ m lm (0)) being the difference between a full matrix element and its FBA result. Here ′ ll ′ has excluded l = l ′ = 0 term. Note that in the limit of a weak external field, we have following orders of magnitudes: a s = O(1), V d = O(D 2 ), and ∆a (m) ll ′ = O(D 4 ). Therefore, it is easy to see that the pseudo-potential, Γ, shown in Eq. (9) has a very smooth connection with the known FBA results [3,4,8,12] in the limit of small dipole moment. From Eqs. (8) and (9), it is straightforward to write down the full effective Hamiltonian to describe the low energy many-body physics of polar molecules: where V ext (p) is the external trapping potential in momentum space. Note that Eq. (11) has included scattering from all channels and is also consistent with the FBA results in the weak dipole limit (|f ∆ | ∝ O(D 4 ) as D → 0). When the external electric field is strong enough, there will be some modification of the scattering amplitude to be beyond the results of first Born approximation even in channels different from s-wave, i.e. ∆a (m) ll ′ = 0 for l, l ′ = 0. Calculating the magnitude of such modification beyond the FBA has to be based on the first principle calculation of two scattering molecules, and is beyond the scope of this work. Our interest in the current paper is to study the effective Hamiltonian and the possible many-body physics when f ∆ is known.
In order to compare with the existing theory of weakly interacting dipoles [3,4,8,12], it is instructive to express Eq. (11) in real space. Details of the transformation is shown in Appendix A. The final result is where ′′ ll ′ has excluded (l, l ′ ) = (0, 0), (0, 2) and (2,0), and we have used the fact that ∆a (m) ll ′ = 0 only for |l − l ′ | = 0, 2, 4, · · · due to the anisotropic nature of dipole potential, V d (r) ∝ Y 20 (r), and its higher order effect. As shown in Appendix A, we have definedφ lm (R) ≡ (l+1)!! 2 l/2 dr Y lm (r) r 3ψ (R + r 2 )ψ(R − r 2 ) as a "pairing" operator in angular momentum (l, m) channel with a spatial "wavefunction" Y lm (r)/r 3 . Although such pairing operators do not represent true composite particles, but can be used to describe the relative motion of two dipoles before and after scattering: the first term of the last line indicates an association-dissociation process between a pair and two dipoles, while the last term describe a transition between "pairs" of different angular momentum channels. These two novel interaction terms should bring complete new physics in a strongly interacting polar molecules, and is worthy for further investigation in the future.
Starting from the effective Hamiltonian, Eq. (12), we can also derive the associated Gross-Pitaeviskii type meanfield equation for condensate dynamics by using i ∂ψ/∂t = [ψ, H] and approximating the bosonic field operator,ψ(r), to be a c-number, Ψ(r). The resulting equation can be written as following form: where Ψ(r) = ψ (r) is the condensate wavefunction. Similarly, one can also derive associated Bogoliubov-de Genne equations for the elementary excitations. We note that the effective Hamiltonian, Eq. (11) and Eq. (12), and meanfield equation, Eq. (13), contain all the effects beyond the simple FBA results, and they will reproduce the known FBA results when taking ∆a (m) ll ′ = 0.

A. Gaussian variational wavefunction
To study the aspect ratio and the stability regime of the condensate profile, it is convenient to use the variational approach [17]. Here we use a Gaussian type trial wavefunction, 14) for the condensate wavefunction in harmonic trapping potential: V ext (r) = 1 2 M ω 2 0 (x 2 +y 2 )+ 1 2 M ω 2 z z 2 , where ω 0 and ω z are the associated trapping frequencies. Here N is the total number of dipoles, and R 0 and R z are the Gaussian radii of the condensate in the x − y plane and along the z axis respectively. The variational energy can be obtained easily from the effective Hamiltonian in the momentum space, Eq. (11), via replacingâ k by Ψ k ≡ â k = 1 √ Ω drΨ(r) e −ik·r . We therefore obtain . β ≡ R z /R 0 and κ ≡ ω 0 /ω z are the condensate and trapping aspect ratios. We have also scaled all the length scales (a s , a d , ∆a (0) ll ′ and R 0 ) by the horizontal oscillator length, a osc,0 ≡ /M ω 0 (i.e.R 0 = R 0 /a osc,0 etc.), and used E 0 ≡ N 2 /2ma 2 osc,0 as the energy scale. The first two terms in the right hand side of Eq. (15) are from the kinetic and potential energies respectively, and the third is from the s-wave scattering channel. The fourth term in from the contribution within the first Born approximation and the second line is from the effects beyond the FBA. Again we find that the whole meanfield energy of Eq. (15) will become the same as calculated within the First Born approximation by taking ∆ã (0) l,l+2 = 0. Using the fact that A 0 (β) = (4β) −1 , we find that the contribution of the FBA is of the same form as the term with ∆ã (0) 0,2 (both of them are proportional to A 2 (β)). However, such coincidence is simply due to the special form Gaussian trial wavefunction. Using other trial wavefunctions can easily give different aspect ratio dependence of these two effects. Besides, we also note that A l (β = 1) = 0 for l = 0, showing that for a spherically symmetric condensate (β = 1), only s-wave scattering channels are relevant: scattering in finite angular momentum channels are cancelled out due to spherical symmetry of the condensate profile. When the condensate profile is highly anisotropic due to external confinement (say in cigar shape, β ≫ 1 or in pancake shape, β ≪ 1), the effects beyond the FBA will become very crucial.

B. Example: near shape resonance
For the general form of effective Hamiltonian of Eqs. (11) and (12), values of a (m) ll ′ have to be obtained from the first principle calculation [8,9], which is however beyond the scope of this paper. In fact, due to the highly nontrivial inter-molecule interaction in short-distance, the low energy scattering matrix element, t l ′ m ′ lm (0), can be very different from the results of Born approximation in strong dipole regime. Here we consider the simplest case to study the effect beyond the FBA: we assume the external electric field is still weak but near the first shape resonance regime, where the shape resonance occurs in the s-wave channel so that both t 00 00 (0) and t 20 00 (0) are strongly deviated from results in the weak interaction limit. Scattering matrix elements in other channels are less affected because of the centrifugal potential for l = 0. This picture is also consistent to the numerical results shown in Ref. [9], where their numerical results of t 40 20 (0) is almost unaffected by the shape resonance in the s-wave channel. (But it does not exclude the possibility to have significant deviation in other channels in the regime of much stronger dipolar interaction.) Under such assumption, we may consider ∆a 0,2 = 0, and ∆ l,l ′ = 0 for all (l, l ′ ) = (0, 2), (2,0) or (0,0). As a result, the variational energy of Eq. (4) becomes (using A 0 (β) = 1/4β): From above result, we find the contribution of the ∆a (0) 0,2 term can reduce (since ∆a (0) 0,2 > 0 near the first shape resonance, see Ref. [9]) the effect of anisotropic feature of dipole interaction. Although this result is derived from the Gaussian trial wavefunction, such reduction of anisotropy of the condensate wavefunction should be still qualitatively correct for the correct condensate profile.
We note that the ground state energy and the pseudo-potential study has been also discussed in Ref. [8], where they include the dipole dependence in the s-wave scattering length (i.e. a s (D)) and use the FBA results (Eq. (2)) for the dipole interaction near the first few shape resonance. In other words, they did not consider the effect of strong deviation of t 20 00 (0) from the FBA, which is a very significant result as shown in Ref. [9]. Therefore, the large value of s-wave scattering length near the shape resonance in Ref. [8] may have smeared out the contribution of ∆a (0) 0,2 . If considering an even stronger dipole moment (larger than the value for the first few shape resonance), where the scattering amplitudes may deviate from the FBA result in all channels, one has to solve Eqs. (11)-(13) with finite values of ∆a (m) l,l ′ for the correct many-body physics of polar molecules.

VI. EFFECTIVE HAMILTONIAN AND EXCITATIONS IN 2D:
In a 2D homogeneous system, we can assume that the wavefunction in the z axis is of Gaussian type: φ(z) = where R z is the width of such quasi-2D potential layer. After integrating out the degree of freedom in z direction (i.e. along the direction of external electric field) of the 3D effective Hamiltonian, Eq. (11), we obtain where we defineb p andb † p to be the field operator in 2D system with p being the in − plane momentum vector from now on. Ω ⊥ is the 2D area, and V s ≡ 4π 2 as √ 2π MRz is the contribution of s-wave scattering.
where g(x) = 1 − (3 √ π/2)x e x 2 Erfc(x) with Erfc(x) being the complementary error function. We also have to account the contribution beyond FBA, where with φ p ≡ tan −1 (p y /p x ) being the angle in 2D plane. At zero temperature, the dipolar atoms/molecules condense at p = 0, so that the total energy is E 2D = , where n 2D = N/Ω ⊥ is the particle density in the 2D plane. Keeping only the condensate part (b 0 =b † 0 = √ n 2D ) and the quadratic order of fluctuations (p = 0), the effective Hamiltonian become: where we have used the fact that V ∆ (p 1 , p 2 ) = V ∆ (−p 1 , −p 2 ) = V ∆ (p 2 , p 1 ). Finally, we could use the Bogoliubov transformation to diagonalize above Hamiltonian and obtain the following phonon excitation spectrum:  where accounts the effects beyond the FBA results. Similar to the 3D case, now we study the situation when only ∆a The calculated dispersion, ω p , for different values of ∆a (0) 0,2 > 0 are shown in Fig. 2. There are two significant effects to be noted: First, in the short wavelength regime, the roton minimum, predicted [4] as a feature of dipolar interaction for 2D bosonic polar molecules, becomes weaker as a (0) 0,2 becomes stronger. Secondly, in the long wavelength limit, instead of the typical linear dispersion [4], we find ω p = C |p|R z (1+ MRz . As a result, the phase fluctuation becomes much stiffer than predicted in the FBA, showing an enhancement of the condensate/superfluid density at zero temperature. More precisely, we can calculate the normal fluid density, ρ n , according to the transverse current correlation function [18]. The sublinear dispersion of ω p gives ρ n (T ) = 7!ζ(7) 2 2πM (kB T ) 7 C 8 R 4 z , which shows a much smaller temperature (T ) dependence than the result obtained for linear dispersion (ρ n = 3ζ (3) if ω p = c 1 |p| [18]). According to Landau's two-fluid model and the universal relation between the 2D superfluid density and the Kosterlitz-Thouless transition temperature (T c ), the superfluid transition temperature (T c ) of 2D dipolar system is then determined by k B T c = π 2 ρ s (T c )/2M = π 2 (n 2D − ρ n (T c ))/2M . At temperature below T c , the single particle correlation function has a power-law decay with zero condensate density. These results are also equivalent to a 2D charged Bose gas (V (r) = Q 2 /r) [10] with an effective charge, Q = C R z M/n 2D 2 . Such an interesting equivalence implies a possibility to use neutral polar molecules to simulate a 2D charged boson system in liquid phase (not doable for ion traps due to the strong Coulomb interaction and large atom mass), which may be important to the understanding of the superconducting Cooper pairs in High T c thin film [11].

VII. SUMMARY
In summary, we have developed a full effective many-body theory for 3D and 2D dipolar Bose gases beyond the simple first Born approximation. One of the significant consequence is that the dipolar interaction effect in the 3D condensate can be reduced near the shape resonance regime. For the 2D system (highly anisotropic regime), such effect brings a significant change of the low energy excitation spectrum. We believe there should be more interesting results for a polar molecule system in strong external field regime, where all scattering channels (besides of the t 20 00 channel) can deviation from the FBA significantly. Our results therefore are especially important for the future studying of the many-body properties of strongly interacting polar molecules.

VIII. ACKNOWLEDGEMENT
We thank G. Baskaran, A. Derevianko, D.-H. Lee, S. Ronen, L. You, and W.-C. Wu for fruitful discussions. Part of this work was done in KITP, Santa Barbara. Our work is supported by NSC Taiwan.