Collective excitations of an imbalanced fermion gas in a 1D optical lattice
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Collective excitations of an imbalanced fermion gas in a 1D optical lattice
R. Mendoza
Posgrado en Ciencias F´ısicas, UNAM; Instituto de F´ısica, UNAM
Mauricio Fortes and M. A. Sol´ıs
Instituto de F´ısica, UNAM, Apdo. Postal 20-364, 01000 M´exico D.F., M´exico
The collective excitations that minimize the Helmholtz free energy of a population-imbalancedmixture of a Li gas loaded in a quasi one-dimensional optical lattice are obtained. These excitationsreveal a rotonic branch after solving the Bethe-Salpeter equation under a generalized random phaseapproximation based on a single-band Hubbard Hamiltonian. The phase diagram describing stabilityregions of Fulde-Farrell-Larkin-Ovchinnikov and Sarma phases is also analyzed.
PACS numbers: 74.70.Tx,74.25.Ha,75.20.HrKeywords: Superfluidity, Bethe-Salpeter, Collective excitations, Phase diagram
I. INTRODUCTION
Optical lattices are tailored made to study strongly-correlated Fermi systems, the stability of different phasesand the effects of dimensionality in population-imbalanced mixtures of different species of ultra cold gases underattractive interactions . In the latter case, the Fermisurfaces of each species are no longer aligned and Cooperpairs have non-zero total momenta 2 q . Such phases werefirst studied by Fulde and Ferrell (FF) , who used anorder parameter that varies as a single plane wave, and byLarkin and Ovchinnikov (LO) , who suggested that theorder parameter is a superposition of two plane waves.The mean-field treatment of the FFLO phase in a vari-ety of systems, such as atomic Fermi gases with popula-tion imbalance loaded in optical lattices , shows thatthe FFLO phase competes with a number of other phases,such as the Sarma ( q = 0) states , but in some regionsof momentum space the FFLO phase is more stable as itprovides the minimum of the mean-field expression of theHelmholtz free energy. In addition, recent calculations onthe FFLO phase of the same system in two- and three-dimensional optical lattices suggest that the regionof stability of this phase as a function of polarization in-creases when the dimensionality of the periodic lattice islowered.In this work, we use a Bethe-Salpeter approach to ob-tain the collective excitations of the two-particle propa-gator of a polarized mixture of two hyperfine states |↑ > and |↓ > of a Li atomic Fermi gas with attractive inter-actions loaded in a quasi one-dimensional optical latticedescribed by a single-band Hubbard Hamiltonian.
II. HUBBARD MODEL
The Hamiltonian of a two-component Fermi gas underan attractive contact interaction in a periodic lattice with constant a is H = − J x X h i,j i x ,σ ˆ c † i,σ ˆ c j,σ − J y X h i,j i y ,σ ˆ c † i,σ ˆ c j,σ − J z X h i,j i z ,σ ˆ c † i,σ ˆ c j,σ − X i (cid:16) µ †↑ ˆ c † i, ↑ ˆ c i, ↑ + µ ↓ ˆ c † i, ↓ ˆ c i, ↓ (cid:17) + U X i ˆ c † i, ↑ ˆ c † i, ↓ ˆ c i, ↓ ˆ c i, ↑ , (1)where J ν is the tunneling strength of the atoms betweennearest-neighbor sites in the ν -direction; U is the on-site attractive interaction strength; µ ↑ , ↓ is the chemi-cal potential of species | ↑ > , | ↓ > , and the Fermi op-erator ˆ c † i,σ (ˆ c i,σ ) creates (destroys) an atom on site i .We assume a system with a total number of atoms M = M ↑ + M ↓ distributed along N sites of the optical-lattice potential. For a quasi one-dimensional (1D) sys-tem the tunneling strengths satisfy J x ≫ J y = J z andthe usual tight-binding lattice dispersion energies are ξ ↑ , ↓ ( k ) = 2 P ν J ν (1 − cos k ν a ) − µ ↑ , ↓ .The order parameter ∆ i = U h ˆ c i, ↓ ˆ c i, ↑ i of the FFLOstates is assumed to vary as a single plane wave, ∆ i =∆ exp (2 ı q · r i ), where 2 q is the pair center-of-mass mo-mentum, r i the coordinate of site i , and ∆ is the usualBCS gap. III. PHASE DIAGRAMS
Within the mean field approximation and using aBogoliubov transformation to diagonalize the Hamil-tonian, the grand canonical partition function Z canbe obtained in terms of both, electronlike and hole-like dispersion ω ± = E q ( k ) ± η q ( k ) , where η q ( k ) = [ ξ ↑ ( k + q ) − ξ ↓ ( q − k )] and E q ( k ) = q χ q ( k ) + ∆ . The thermodynamic potential Ω = − β ln Z isΩ = 1 N X k (cid:20) χ q ( k ) + ω − ( k , q ) + ∆ U (cid:21) − β X k h ln (cid:16) e − βω + ( k , q ) ) + ln(1 + e βω − ( k , q ) (cid:17)i , (2)where χ q ( k ) = [ ξ ↑ ( k + q ) + ξ ↓ ( q − k )] and β =1 /k B T. The Helmholtz free energy F (∆ , q , f ↑ , f ↓ , T ) =Ω + µ ↑ f ↑ + µ ↓ f ↓ , where f ↑ , ↓ ≡ M ↑ , ↓ /N is the spin-up(spin-down) filling fraction, can now be minimized withrespect to ∆ , q , µ ↑ and µ ↓ . This leads to a set of fourequations that determine the stable phases of this systemas a function of temperature and polarization P ≡ f ↑ − f ↓ f ↑ + f ↓ .f ↑ = 1 N X k (cid:2) u q ( k ) f ( ω + ( k , q )) + v q ( k ) f ( − ω − ( k , q )) (cid:3) ,f ↓ = 1 N X k (cid:2) u q ( k ) f ( ω − ( k , q )) + v q ( k ) f ( − ω + ( k , q )) (cid:3) , UN X k − f ( ω − ( k , q )) − f ( ω + ( k , q ))2 E q ( k )0 = 1 N X k (cid:26) ∂η q ( k ) ∂q x [ f ( ω + ( k , q )) − f ( ω − ( k , q ))] + ∂χ q ( k ) ∂q x × (cid:20) − χ q ( k ) E q ( k ) [1 − f ( ω + ( k , q )) − f ( ω − ( k , q ))] (cid:21)(cid:27) , (3)where u q ( k ) = r h χ q ( k ) E q ( k ) i , v q ( k ) = r h − χ q ( k ) E q ( k ) i and f ( x ) is the Fermi distribution function. U/J x = T = -4 E R f = P µ ↑ / E R µ ↓ / E R q x a / π ∆ / E R FIG. 1: Gap (dots), chemical potentials and pair momentumthat minimize the Helmholtz free energy.
In Fig. 1 we show the gap ∆, pair momentum q x andchemical potentials for each species µ ↑ , µ ↓ as a function of polarization that minimize the Helmholtz free energyfor a total filling factor f = f ↑ + f ↓ = 0 . J x = 0 . E R , J y = J z = 10 − E R , and U/Jx = 2 .
64, where E R = ~ (2 π/λ ) / m is the recoil en-ergy and λ = 1030 nm is the lattice wavelength. When P = 0 , q x = 0 and ∆ = 0 the system is in the FFLOphase which becomes the BCS state when both P → q x →
0; when P = 0 , but q x = 0 the system is in theSarma or breached-pair phase and the transition tothe normal state occurs when the gap ∆ vanishes. J x = 0.078 E R , f = 0.4685, U/J x = 2.64 P T/E R FFLO S FIG. 2: (Color online) Phase diagram of an imbalanced Fermigas in a quasi 1D optical lattice.
The phase diagram of the quasi 1D system is shown inFig 2. It is interesting to note that the FFLO phase isdominant over quite a large region of the phase-diagramand is stable at higher values for the polarization (up to P ≃ .
64) compared to our previous work in 2D and3D systems with the same composition and dynamicalparameters. In addition, the mixed phase or phase sep-aration regime in which an unpolarized BCS core and apolarized normal fluid (in momentum space) coexist atvery low temperatures and moderate polarizations isno longer present in this system in contrast to the 3Dsystem (and to a lesser extent in the 2D optical lattice). IV. COLLECTIVE STATES
The spectrum of the collective modes can be ob-tained from the poles of the two-particle Green’s func-tion K (1 ,
2; 3 , , where we use the compact notation1 = { σ , r , t } , { σ , r , t } , ... with σ i denoting thespin variables, r i the vector for lattice site i , and t i , thetime variable. K satisfies the following Dyson equation: K = K + K IK, (4)where K (1 ,
2; 3 ,
4) is the two-particle free propagatorwhich is defined by a pair of fully dressed single-particleGreen´s function, K (1 ,
2; 3 ,
4) = G (1; 3) G (4; 2) , and the interaction kernel I is given by functional deriva-tives of the mass operator.Using the generalized random phase approximation,we replace the single-particle excitations with those ob-tained by diagonalizing the Hartree-Fock (HF) Hamilto-nian while the collective modes are obtained by solvingthe Bethe-Salpeter (BS) equation in which the single-particle Green’s functions are calculated in the HF ap-proximation, and the BS kernel is obtained by summingladder and bubble diagrams . The resulting equationfor the BS amplitudes ˆΨ q ( k, Q ) isˆΨ q ( k, Q ) = − U ˆ D X p ˆΨ q ( p, Q ) + U ˆ M X p ˆΨ q ( p, Q ) , (5)where ˆΨ q ( k, Q ) is a vector with four components andthe 4 × U ˆ D and U ˆ M represent the contribu-tion resulting from the direct and exchange interactions,respectively. The dispersion ω ( Q ) for the collective exci-tations is obtained from the solutions of the 4 × E xc i t a t i on E ne r g y ( E R ) Q x ( π / a ) sound mode FIG. 3: Excitation energy for collective modes of a polarized Li gas in a quasi 1D optical lattice with λ = 1030 nm andtotal filling factor f = 0 . . The Hubbard parameters are J x = 0 . E R , J y = J z = 10 − E R and the attractive on-siteinteraction is U/J = 2 . . lective excitations of the 1D system. For small Q theGoldstone mode is clearly present with a sound velocityof v s = 4 .
92 mm/s. For larger Q , a rotonlike minimumappears with a gap ∆ r = 0 . E R and a critical flowvelocity v f = 1 .
05 mm/s.
V. CONCLUSIONS
We have shown that superfluid phases of the FFLOand Sarma types are present in ultra cold Fermi gasesloaded in quasi one-dimensional optical lattices. The re-gion of stability of the FFLO states in the phase diagramis larger and supports higher population imbalances thanidentical systems in 2D and 3D. The energy dispersion ofcollective excitations have the usual Goldstone-mode be-havior with a sound velocity of 4.92 mm/s. In addition,for higher momenta a rotonic branch is also present.
Acknowledgments
We acknowledge the partial support from UNAM-DGAPA grants IN105011, IN-111613 and CONACyTgrant 104917. T. Esslinger, Ann. Rev. Condensed Matter Phys. , 129(2010). Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle,Nature , 689 (2008). W. Ketterle, Y. Shin, A. Schirotzek and C. H. Schunk, J.Phys, Condensed Matter , 164206 (2009). Yean-an Liao, A.S.C. Rittner, T. Paprotta, W. Li, G.B.Partridge, R.G. Hulet, S.K. Baur and E.J. Mueller, Nature , 567 (2010). P. Fulde, and R. A. Ferrell, Phys. Rev. , A550 (1964). A. I. Larkin, and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz., , 1136 (1964) [Sov. Phys. JETP , 762 (1965)]. T. Koponen et al. , New Journal of Physics , 179 (2006) T. Koponen et al. , Phys. Rev. Lett. , 120403 (2007);T. Paananen, T. K. Koponen, P. T¨orma, and J.P. Mar-tikainen, Phys. Rev. A , 053602, (2008). Tung-Lam Dao, A. Georges, and M. Capone, Phys. Rev.B , 104517 (2007); Q. Chen et al. , Phys. Rev. B , , 043605 (2007); M. Rizzi, et al. , Phys.Rev. B , 245105 (2008); Xia-Ji Liu, Hui Hu, and P.D. Drummond, Phys. Rev. A , 023601 (2008); M. RezaBakhtiari, M. J. Leskinen, and P. T¨orma, Phys. Rev. Lett. , 120404 (2008); A. Lazarides and B. Van Schaeybroec,Phys. Rev. A , 041602 (2008); T Paananen, J. Phys.B: At. Mol. Opt. Phys. , 180509(R) (2009); A. Mishraand H. Mishra, Eur. Phys. J. D , 75 (2009); B. Wang,Han-Dong Chen, and S. Das Sarma, Phys. Rev. A ,051604(R) (2009); Y. Yanase, Phys. Rev. B , 220510(R)(2009); A. Ptok, M. M´aska, and M. Mierzejewski, J. Phys.:Condens. Matter , 295601 (2009); Yan Chen et al. , Phys.Rev. B , 054512 (2009); Yen Lee Loh and N. Trivedi,Phys. Rev. Lett. , 165302 (2010); A. Korolyuk, F. Mas-sel, and P. T¨orma, Phys. Rev. Lett. , 236402 (2010); F.Heidrich-Meisner et al. , Phys. Rev. A , 023629 (2010);S. K. Baur, J. Shumway, and E. J. Mueller, Phys. Rev. A , 033628 (2010); A. Korolyuk, F. Massel, and P. T¨orm¨a,Phys. Rev. Lett. , 236402 (2010); M. J. Wolak et al. ,Phys. Rev. A , 013614 (2010); L. Radzihovsky and D.Sheehy, Rep. Prog. Phys. , 076501 (2010). K. Machida, T. Mizushima and M. Ichioka, Phys. Rev. Lett. T.L. Dao, M. Ferrero, A. Georges, M. Capone and O. Par-collet, Phys. Rev. Lett. Z. G. Koinov, R. Mendoza and M. Fortes, Phys. Rev. Lett. , 100402 (2011). R. Mendoza, M. Fortes, M.A. Sol´ıs and Z.G. Koinov,arXiv:1306.4706 (2013). G. Sarma, J. Phys. Chem. , 1029 (1963). W.V. Liu and F. Wilczek, Phys. Rev. Lett., , 047002(2003). P. F. Bedaque, H. Caldas, and G. Kupak, Phys. Rev. Lett. , 247002 (2003); H. Caldas, Phys Rev. A , 045008 (2005); S. Sachdev and K. Yang, Phys.Rev. B , 174504 (2006).(1994). Z. G. Koinov, Physica C , 470 (2010); Physica StatusSolidi (B) , 140 (2010); Ann. Phys. (Berlin) , 693(2010); cond-mat/1010.1200. K. V. Samokhin, Phys. Rev. B , 224507 (2010). Y.-P. Shim, R. A. Duine, and A. H. MacDonald, Phys.Rev. A74