Induced Interactions for Ultra-Cold Fermi Gases in Optical Lattices!

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1 D.-H. Kim, P. Törmä, and J.-P. Martikainen, arxiv: , PRL, accepted. Induced Interactions for Ultra-Cold Fermi Gases in Optical Lattices! Dong-Hee Kim 1, Päivi Törmä 1, Jani-Petri Martikainen! 1, Finland NORDITA, Sweden

2 What we want to study:! Superfluid transition of Fermi gases (Two component) (Attractive interaction) in Optical Lattices within the Mean-Field Theory (3D, D, ) (weak coupling regime) beyond BCS predictions (many-body effect) Medium-Induced Interaction : Gorkov Melik-Barkhudarov (GMB) correction

3 Medium effect in FREE gases! BCS result in vacuum : k B T () c = 8γ πe ɛ F exp ( π ) k F a U = 4π m a Sá de Melo et al., PRL 71, 3 (1993). Stoof et al., PRL 76, 1 (1996). Correction due to a Medium effect: Gorkov Melik-Barkhudarov (GMB) correction T () c T c = (4e) 1/3. : nd order correction to the interaction. Gorgov and Melik-Barkhudarov, Sov. Phys. JETP 13, 118 (1961). Heiselberg et al., PRL 85, 418 ().

4 Superfluidity in optical lattices?! Indirect experimental evidence: J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C. Sanner, K. Xu, and W. Ketterle, Nature 443, 961 (6): ~ long range order? Q. Medium effects in optical lattices? The difference could be even more dramatic. ( T c exp π ) k F a : correction changes the prefactor. k F a k F a (1 + δk F a ) : nd order correction to interaction T c e δ T c : varies sensitively with the correction

5 Gorkov Melik-Barkhudarov correction! Physical interpretation p p + k + q k H. Heiselberg, C. H. Pethick, H. Smith, and L. Viverit, PRL 85, 418 (). C. J. Pethick and H. Smith 8. Two fermions in the medium two fermions in vacuo. Scattering in the medium: p q medium-induced interaction k q k p + k + q p p p + k + q k q

6 Induced Interaction calculation! Induced interaction: U ind (p, k) = U dq (π) D Effective interaction: U ind = f,p+k+q f,q ξ (p + k + q) ξ (q) f σ,k = Assume only the momenta at the Fermi surfaces contribute: 1 S S S ds p S ds k U ind (p, k) ξ σ (k) =ɛ σ (k) µ σ exp [βξ σ (k)] U eff = U + U ind : effective interaction replacing U TWO INTERESTING FEATURES

7 Induced Interaction: continued! 1. Lindhard function (static) susceptibility for non-interacting case q U ind (p, k) = U p + k + q dq (π) D f,p+k+q f,q ξ (p + k + q) ξ (q) POSITIVE : screens the negative interatomic interaction U suppresses T C or the order parameter.. Perfect screening occurs at large interactions. U eff = U c + U ind = - parameter space of validity At this large induced interaction, a RPA charge susceptibility can diverge. χ RP A = χ 1+U χ - connection to other physics. (Charge density waves)

8 Mean-Field calculations! Hubbard Hamiltonian: H = σα t σα (ĉ σ,i α +1ĉσ,i α + h.c.)+u ˆn iˆn i µ σ,i ˆn σi i α Mean-Field approximation: = U ĉ iĉ i i α {x, y, z} (ξ k ĉ kĉ k + ξ k ĉ kĉ k + ĉ kĉ k + ĉ kĉ k ) H MF = k U U U : the usual BCS calculations ɛ σ (k) = α t σα [1 cos(k α )] Energy dispersion of non-interacting system U U eff : the effective interaction with the GMB correction. The zero temperature order parameter Δ is calculated.

9 Mean-Field calculations! Minimization of Free Energy: T.K. Koponen, T. Paananen, J.-P. Martikainen, and P. Törmä, PRL 99, 143 (7). T.K. Koponen, T. Paananen, J.-P. Martikainen, M.R. Bakhtiari, and P. Törmä, NJP 1, 4514 (8). Ω = U + dk (π) D ( ξ ( k)+e (k) 1 ) β ln(1 + e βe +(k) )(1 + e βe (k) ) Effective interaction U eff = U + U ind E ± (k) = ξ (k) ξ ( k) (ξ ) (k)+ξ ( k) ± + µ = µ : Perfectly balanced gases are considered. Numerical Integrations: Monte Carlo algorithm is used.

10 versus T C! The zero temperature order parameter Δ is calculated instead of T C D D!/k B T C = 3.53! / k B T C ! /! half-filling Both are equivalent in the weak coupling regime.

11 3D Lattices: Induced Interaction! Induced interaction strength increases as chemical potential increases. U ind (p, k) = U dq (π) D f,p+k+q f,q ξ (p + k + q) ξ (q) U eff decreases as µ increases.!u ind " [t] (U = 3t, T = ) ! [t] U [t] 8 U c 6 4 U U eff 4 6 µ [t] U < U c for all µ. Half-filling

12 3D Lattices: order parameter (T=)! The order parameter is suppressed much beyond the factor..! [t] U c 6 4 U U eff 4 6 BCS As the filling factor increases, correction effect becomes larger. BCS GMB. BCS GMB 5 at low density limit. at µ = 4t ! [t] (U = 3t, T = ) 3D van hove singularity GMB half filling BCS GMB 5 cf.) 1/D correction at high dimension [van Dongen, PRL 67, 757 (1991)] T c /T c O(1) at half filling.

13 D Lattices: induced interaction! Induced interaction strength increases as chemical potential increases, but induced interaction diverges at half filling.!u ind " [t] U ind (p, k) = U U [t] 4 U eff U 4 µ [t] dq (π) D U c 1 3 4! [t] f,p+k+q f,q ξ (p + k + q) ξ (q) Half-filling!U ind " [t] 1.!U ind " ~ ln(4-!) ! [t] (U = 1.5t, T = )

14 D Lattices: order parameter (T=)! As µ increases, correction effect becomes larger U U c BCS - Effect of GMB correction BCS GMB 7 at µ = 3t. - Comparisons with QMC 1. Decrease near half filling! [t].1 U eff 4 [PRL 6, 147 (1989)].5 GMB 1 3 4! [t] (U = 1.5t, T = ). Quantitative comparison At quarter filling and U =-4t, k B T c.5t [PRL 6, 147 (1989); PRL 66, 946 (1991)] GMB correction: k B T c.t

15 D Lattices : divergence at half filling!!u ind " [t] !=4.!=3.995!=3.99!=3.98 U ind (p, k) = U The induced interaction diverges at half filling at T=. q dq (π) D p + k + q f,p+k+q f,q ξ (p + k + q) ξ (q) p + k :constant /k B T [t -1 ]!! (-!,!) (!,!) Fermi surface nesting: ex. 1D: p + k k F k y -! (-!,-!) (!,-!) Divergence of the Lindhard function signature of CDW -! -! -!!! k x Superfluidity and CDW coexist in the ground state at half filling in D lattices.

16 Lattice Anisotropy: crossover from 3D to 1D! t = 1D 3 t =1 3D Lattice anisotropy: 1D limit: t y,z t = t y /t x = t z /t x!u ind " [t x ] Induced Interaction increases as the lattice goes toward quasi 1D. Kinks are found at two points of changes in Fermi surface shapes. t =.5 The surface opens: van Hove singularity.5..4 ~ U = 3t µ =t t # t y / t x = t z / t x t =.5 Quasi 1D shapes start to develop. t : Fermi surface nesting occurs. Correction diverges.

17 Lattice Anisotropy: crossover from 3D to 1D!! [t x ] t = 1D D U = 3t µ =t BCS GMB ~ t t =1 The order parameter decreases much beyond the BCS prediction. However, the theory fails at 1D limit. U ind diverges in 1D lattices. U eff = at some point: ( incorrect! ) In quasi 1D lattices, finite gap exists. (Larkin and Sak, PRL 1977; PRB 1978.)

18 Summary! D.-H. Kim, P. Törmä, and J.-P. Martikainen, arxiv: , accepted in PRL. 5! BCS /! corr D Lattice D Lattice corr BCS The presence of the optical lattice significantly enhances the effect of induced interactions on the BCS superfluidity. µ / µ half-filling The induced interaction correction extends the applicability of the mean-field calculations in lower dimensions. - Predictions closer to QMC values in D lattices - Divergence due to Fermi surface nesting : connection to different physics