Boson-Fermion pairing in Bose-Fermi mixtures on 1D optical lattices
aa r X i v : . [ c ond - m a t . o t h e r] N ov Boson-Fermion pairing in Bose-Fermi mixtures on 1D optical lattices
X. Barillier-Pertuisel , S. Pittel L.Pollet P. Schuck , , Institut de Physique Nucl´eaire, IN2P3-CNRS, UMR8608, Orsay,F-91406, France Universit´e Paris-Sud,F-91406 Orsay, France Bartol Research Institute and Department of Physics andAstronomy, University of Delaware, Newark, Delaware 19716, USA Institut Theoretische Physik, ETH Z¨urich, CH-8093 Z¨urich, Switzerland Laboratoire de Physique et Mod´elisation des Milieux Condens´es, CNRS Universit´eJoseph Fourier, Maison des Magist`eres, B.P. 166, 38042 Grenoble Cedex 9, France (Dated: November 7, 2018)Boson-fermion pairing is considered in a discrete environment of bosons and fully spin-polarizedfermions, coupled via an attractive Bose-Fermi Hubbard Hamiltonian in one dimension. The resultsof the T-matrix approximation for particles of equal mass and at double half filling are comparedwith the results of exact diagonalization and with Quantum Monte Carlo results. Satisfactoryagreement for most quantities is found. The appearance of a stable, weak-coupling pairing mode isalso confirmed.
PACS numbers: 03.75.Fi, 05.30.Fk
One of the most intriguing aspects in the field ofcold atoms involves the study of mixed Bose-Fermi(B-F) systems. Several boson-fermion mixtures havebeen realized, both in [1] and without [2] an opticallattice, and their properties have been studied. Sev-eral very interesting phenomena unique to mixed BFsystems have been predicted, including for examplethe possibility of forming composite fermions throughthe pairing of a boson and a fermion [3, 4]. Onthe theoretical side, B-F mixtures have been studiedin mean-field approximation [5, 6] and with variousmethods for treating B-F mixtures in one dimension(1D). This includes exact solution using the Betheansatz [7], bosonization techniques [8] and a Quan-tum Monte Carlo treatment for a B-F mixture on anoptical lattice [9]. There remains, however, the needfor theoretical approaches capable of reliably describ-ing B-F mixtures in more than one dimension [10].A possible theoretical approach to mixed boson-fermion systems was proposed recently in the con-text of nuclear physics, to provide a framework fordescribing the transition from a fermi gas (of quarks)to one of composite fermions (nucleons), i.e. boundthree-quark states. This problem was simplified byassuming that two of the quarks are strongly boundand form a boson. In this way, the extremely complexin-medium three-body problem [11] was replaced bythe much simpler two-body problem of fermion-bosonscattering, for which a T matrix approach was devel-oped [12]. An interesting result of that study was that,due to the presence of a Fermi surface, a stable B-Fbranch was created for an arbitrarily small B-F at-traction. The underlying mechanism turned out to beanalogous to the formation of stable Cooper pairs in apure fermi-gas, though in the latter case the pairs areboson-like, whereas here they are fermion-like. Sev- eral interesting questions follow naturally. On the onehand, we would like to know how reliable the informa-tion provided by the T-matrix approach developed in[12] is when dealing with complex systems involvingbosons and fermions. If it is found to be acceptablyreliable, we might then hope to further develop themethod for application to the variety of systems inwhich boson and fermions degrees of freedom coexist,such as those that arise in cold atomic gases.As the next step in this program, we report in thiswork a study of B-F pairing in 1D optical latticesusing the above T-matrix approach. Such systemscan be treated stastistically exactly for a fairly largenumber of bosons and fermions using Quantum MonteCarlo methods [9], thereby providing an appropriatetesting ground for our method. As we will see, eventhough 1D is a quite unfavorable case for the appli-cability of a T-matrix approach in ladder approxima-tion, most quantities are nevertheless reproduced rea-sonably well with this approach over a large range ofcoupling strengths.To be more specific, we consider the bosons andfermions on a 1D lattice governed by a Hubbard modelhamiltonian. The B-F interaction is assumed to beattractive and the B-B interaction to be repulsive. Weassume further that there are no interaction amongfermions, although a repulsive F-F interaction shouldnot alter qualitatively our conclusions. The Hubbardmodel hamiltonian for such a system is given by H = − t b L X
4. The range of U bf values is arbitrary. However,since we are interested in B-F pairing, the range of U bb values is, in principle, restricted to U bb ≥ | U bf | / U bb lead to phase separation, i.e. theB-F pairs cluster together and occupy only half of theavailable space [9]. On the other hand, our T-matrixapproximation does not allow U bb -values greater than U bb ≃ | U bf | , since U bb is only purely treated in ourtheory, i.e. only in HF approximation. Improving onthis point is possible but left for the future. We there-fore limit ourselves to | U bf | ≥ U bb ≥ | U bf | /
2. As anintermediate value we take U bb = | U bf | throughout. |U bf | E x c i t a t i o n e n er g i e s ExactRPA (a) |U bf | E x c i t a t i o n e n er g i e s ExactRPA (b)
FIG. 1: (Color online) Excitation energy as a functionof | U bf | for the 6-site B-F Hubbard model and K = π/ K = 0 (b). The solid lines correspond to theresults of exact diagonalization and the dashed lines to theRPA results. In Fig. 1, we present some typical examples of ex-cited states. We see that the agreement between ap-proximate and exact excitation energies is quite satis-factory. We should note that we have chosen exampleswhere in the exact case there are only low degenera-cies at U bf = 0, since RPA, because of its very lowdimension, can not well reproduce a high degeneracyof uncorrelated configurations, even if there are alsocases in RPA where in the uncorrelated limit degen-eracies occur. However these degeneracies are alwaysless numerous than in the exact case. This is naturalbecause of the dramatically reduced size of the RPAmatrices with respect to the size of the exact ones. Itshould nevertheless be noted that the RPA excitationenergies somehow represents the average trend of thebunch of exact levels.So far we have not invested effort to exactly diago-nalize problems with higher numbers of sites, since thedimensions of the matrices grow exponentially. How-ever, exact QMC results for ground state propertiesfor higher number of sites are available [9]. For L = 70we show a comparison of the exact ground state energy E with the RPA in Fig. 2 (see [12] for the expressionfor E ). The error bars for QMC are smaller than thepoint size. Again the agreement is reasonable up torather high values of | U bf | . |U bf | -12-10-8-6-4-20 E / L QMCRPAHF
FIG. 2: (Color online) Ground state energy per site as afunction of | U bf | for L = 70. The solid line correspondsto the QMC, dotted line to the HF approximation anddashed line to the RPA. We also calculated the occupation numbers n bf ( K ) = P kk ′ < c + K − k b + k b k ′ c K − k ′ > of the B-F pairsand compare them with the exact results in Fig. 3a.In our case the B-F occupation numbers are simplyobtained from the residue of the B-F Green’s function(5) at the poles E ρ . We see that the agreement withthe exact result is quite satisfactory. One might won-der about the upward tendency around K = π/ n f ( k ) = < c + k c k > can only be obtained in a somewhat indirect waywithin our formalism. This goes, however, completelyparallel to what is known from the usual RPA formal-ism for fermions [16]. The corresponding expressionis given by n f ( k ) = X ρ | X ρ k | Θ( ε F − ε f k ) + X K ,ρ | Y ρ k , K − k | Θ( ε f k − ε F )(9)One can demonstrate that (9) conserves fermion par-ticle number. In Fig. 3b the fermion occupation num-bers are shown for two cases of the coupling U bf = − , −
6. Again the agreement with the exact case israther good, in spite of the fact that for U bf = − n b f ( K ) QMC (U bf =-6)RPA (U bf =-6) π/2 π K (a) n f ( k ) QMC (U bf =-2)RPA (U bf =-2)QMC (U bf =-6)RPA (U bf =-6) π/2 π k (b) FIG. 3: (Color online) (a)BF occupation numbers as afunction of the total momentum K , and (b)Fermion occu-pation numbers as a function of the relative momentum k ( U bb = 0 . | U bf | , L = 70). of particles condensed into the q = 0 state whereas theexact solution is, of course, totally distributed withno particles in the condensate. In spite of this fail-ure, we conclude, however, that our first objective ofthe work has been realized, namely our T-matrix ap-proach which has been applied for the first time to thein-medium B-F case in [12] seems to work reasonablywell. In addition, for t b = t f , our formalism is stillvalid. We compare in Table I the ground-state ener-gies obtained by RPA, HF, and QMC for t f = 4 t b , andsee that the results are still of the same quality as forthe case with equal tunneling amplitude (see Fig. 2).Similar conclusions hold for the occupation numbersand for a comparison with exact diagonalization for asystem size L = 6. We also should note that we even- ( U bf ; U bb ) E RPA E HF E QMC (-40;6) -24.30 -12.85 -20.78(-50;10) -29.71 -14.88 -25.54(-60;12) -35.69 -17.15 -30.33TABLE I: Comparison of the ground state energies persite obtained by RPA, HF and QMC for a system of size L = 30 and tunneling amplitudes t f = 4 t b . The HF andRPA relatives errors are roughly constant. tually would like to apply our theory to the 3D casewhere mean field and RPA theories usually performmuch better, and where no QMC data are available.Another objective of this work is to confirm a sur-prising finding discussed in [12], the appearance of a second stable B-F branch at arbitrary small B-F(attractive) interaction. It exists only because of thepresence of a sharp Fermi surface and is thus analo-gous to the formation of Cooper pairs in pure Fermisystems. For this investigation, we pass to the con-tinuum limit in which the B-F propagator in HF ap-proximation is given analytically by G K ( E ) = xπLP √ x − (cid:16) arctan (cid:16) ( x + 1) cot( K + k f ) √ x − (cid:17) (10) − arctan (cid:16) ( x + 1) cot( k f − K ) √ x − (cid:17)(cid:17) + n /LP + 2 + 2 cos ( K ) , with x = P K ) ,P = E + 2 k f π ( U bb + U bf ) + iη ,P = E + k f π ( U bb + U bf ) + iη . In Fig.4 we show the spectral function, i.e. − Im ( G ( K, E )) /π , for a typical set of parameters for N B = N F = L/
4, away from half filling. Besides thelow lying peak which corresponds to the free fermiondispersion in the U bf → U bf . It is the same phe-nomenon as was seen in our earlier work [12]. Theexistence of a Fermi surface entails a logarithmic di-vergence of Re ( G ) at E/ ε F and then there isalways a sharp state below the continuum solutionof 1 − g bf G ( K , E ) = 0. However contrary to whathappens in the homogeneous continuum case [12], thissecond branch does not interact strongly with the low-est free fermion-like branch. In [12] there was a levelcrossing of the two branches which does not occurhere. Also the upper branch has rather little spec-tral weight compared with the lower one. It wouldbe interesting to see whether this second branch canbe found experimentally. In 1D, however, we shouldnote that this slightly detached second branch is cer-tainly an artefact of the T-matrix approximation be-cause the correlated fermion occupation numbers donot show any discontinuity (see fig. 3). However in 2Dor 3D, we think that this second branch should exist.Whether it survives in a trap geometry [17] remainsto be seen.In conclusion, we have investigated in this workboson-fermion pairing in a Bose-Fermi mixture on a1D optical lattice. The in-medium Boson-Fermionscattering problem was solved in T-matrix approx-imation. As in a previous investigation, in thethermodynamical limit, two stable branches were -8-4048 p o l e d i s p er s i o n π/2 π K (a) -10 -5 0 5 10 E - I m ( G ( K , E )) / π (b) FIG. 4: (Color online) (a) The solid line with circles represents the free fermion-like branch of the spectral function, thesolid line the second B-F branch, the solid lines with squares the limits of the continuum and the dashed line a secondplateau in the continuum, (b) The spectral representation corresponding to (5) for K = π/
4, see vertical broken line of(a) ( U bf = − k F = π/ found. One corresponds to the elastic scatteringof the fermions off the Bose condensate and theother is created from scattering of bosons out of thecondensate with fermions above the Fermi sea. Thelatter comes because of the presence of a sharp Fermisurface and therefore has the same origin as theCooper pole in a pure two component Fermi system.While in 1D systems, the second branch is unphysicaland just an artefact of the method, we think that in3D such a branch should be real. We checked thevalidity of our approach versus exact results availablefor a finite number of sites. For most quantitieswe found good qualitative and semi-quantitativeagreement. This is satisfying because, as already mentioned, RPA generally works better in higherdimensions. More elaborate studies of B-F pairingare under way.Ongoing collaboration and discussions with T.Suzuki and C. Martin on B-F correlations are appre-ciated. We gratefully acknowledge contributions byJ. Dukelsky during the early stages of this work, thefinancial support of the Swiss National Science Foun-dation and the partial support of the US NationalScience Foundation under grant [1] K. G¨unter et al. , Phys. Rev. Lett. , 180402 (2006);S. Ospelkaus et al. , Phys. Rev. Lett. , 180403(2006).[2] A. G. Truscott et al. , Science , 2570 (2001); Z.Hadzibabic et al. Phys. Rev. Lett. , 160401 (2003);G. Roati et al. , Phys. Rev. Lett. , 150403 (2002).[3] A. B. Kuklov and B. V. Svistunov, Phys. Rev. Lett. , 100401 (2003).[4] M. Lewenstein, L. Santos, M. A. Baranov and H.Fehrmann, Phys. Rev. Lett. , 050401 (2004).[5] A. Albus, F. Illuminati and J. Eisert, Phys. Rev. A , 023606 (2003).[6] H. Fehrmann, M. A. Baranov, B. Danski, M. Lewen-stein and L. Santos, Optics Communications , 23(2004).[7] A. Imambekov and E. Demler, Phys. Rev. A ,021602 (2006)[8] L. Mathey et al. , Phys. Rev. Lett. , 120404 (2004);L. Mathey Phys. Rev. B , 144510 (2007). [9] L. Pollet, M. Troyer, K. Van Houcke and S. M. A.Rombouts, Phys. Rev. Lett. , 190402 (2006).[10] H.P. B¨uchler and G. Blatter, Phys. Rev. Lett. ,130404 (2003).[11] A. Rapp, G. Zar´and, C. Honerkamp and W. Hofstet-ter, Phys. Rev. Lett. , 160405 (2007).[12] A. Storozhenko, P. Schuck, T. Suzuki, H. Yabu, J.Dukelsky, Phys. Rev. A , 063617 (2005).[13] A.L. Fetter, J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971).[14] P. Ring, P. Schuck,
The Nuclear Many Body Problem (Spinger, New York, 1980).[15] I. Titvinidze, M. Snoek, W. Hofstetter, cond-mat/0708.3241 (2007)[16] A. Bouyssy, N. Vinh Mau,
Nucl. Phys. A229