arXiv:cond-mat/0702660v1 [cond-mat.supr-con] 27 Feb 2007 iu tde)wsapidol o h regime the for pre- only all applied in was (as of studies) but type vious this limitations study mean-field to beyond approach system of first experimen- methodology the our for on was based reference analysis[11], work, recent useful more quantitative a A a interplay be talists. in delicate can effects the which correlation manner capture of to competition unable and phase- are rich the but of ap- picture space these general expected, good As a 10]. provide 9, proaches 8, 7, both bosons[6, for and approaches numbers fermions occupation of fluctuating by for type allowing system variational and two-species or this mean-field with employing associated space phase condensed of phenomena. theoreti- understanding matter of our a section furthering provide large physics[5] a cal can for particular, system ground testing In (BFM) fascinating mixture both alike. boson-fermion playground theorists sys- a physics and of favorite wide experimentalists type a a for these them over with renders control associated tems experimental parameters of Berezinskii– The in- range few the a ones. name and dicative to just gas[3], transition[4], Mott-insulator Tonks-Girardeau Kosterlitz-Thouless realizati the the the of of superfluidity[2], realization fermionic the phe- transition[1], physics as matter such condensed modern span- nomena of results variety remarkable a produced ning have lattices optical c ittdb h em velocity Fermi the dynam- by fermionic dictated the ics between effects retardation where yaisdcae ytesudvelocity sound the by dictated dynamics n opeeto h hscludrtnigo tby it of understanding physical system the of type on this complement on role attention and have our crucial considerations focus above a to The us play prompted orders. can of interplay retardation the fact, in in and cessible otn.I svr motn oivsiaetephysics the investigate since to regime important this very beyond is It portant. rvoswr a xlrdpr ftevs n rich and vast the of part explored has work Previous in atoms ultracold manipulating in advances Recent arn n est-aepae nBsnFrinmixtures Boson-Fermion in phases density-wave and Pairing ASnmes 03.75.Mn,05.10.Cc,71.10.Fd,71.10.Hf numbers: it PACS how and order, dominant the problem. with the associated kee resp gap by calculate energy account We the into self-energies. taken and fully vertices fun interaction are use We effects retardation phase. wave (half- antiferromagnewhere density charge site undergo a to per or superconductivity system one fermion the (for of drives strengths density interactions) interaction fermion c and fixed in parameters at coupling lattice system a electron-phonon atoms the this fermionic with for the similarities between has st coupling BEC (isospin) The hyperfine two (BEC). in condensed prepared are fermions the where esuyamxueo emoi n ooi odaoso two a on atoms cold bosonic and fermionic of mixture a study We v hsc eatet nvriyo aiona Riverside, California, of University Department, Physics F ≥ v s seprmnal ac- experimentally is v F .D Klironomos D. F. n h bosonic the and v s r o im- not are Dtd eray3 2008) 3, February (Dated: v F ≪ v on s , ∗ n .W Tsai S.-W. and ne-pce (fermion-boson inter-species osssol fhrcr ost)itaseis(fermion intra-species approximation (onsite) fermion hardcore good of a only forces consists Waals der van short-range ae al’ xlso rnil iiain o s-wave for limitations principle two- exclusion a re- Pauli’s fermions of among laxes presence number quantum the isospin Additionally, component interactions. tive o emosand fermions for n ooso atc steHbadmdl o a ( For atoms neutral model. Hubbard of the consisting is system lattice BFM a typical on bosons context. and that within valid remain All that will results mode interaction. our collective a fermionic of gapless of the a attractively type square of renormalizes any presence a the for on in generalized system lattice be fermionic realize can itinerant to study interesting two-component our is retardation What that taking is account. inter- while into of undergo, fully orders can effects possible fermions the all the investigate an est apply build to consequently and analysis and out fRG theory, bosons the fermionic integrate field)[11]. effective bosonic to the us in allows order This (BEC) first fermion-boson to phase of (linear type system condensation interaction electron-phonon bosonic an Bose-Einstein the to the leads is which work in to that be study ingredient to this necessary for only required The is presupposes order. always manifested which all to example a contrary for that basis, theory equal so field an way, mean on studied unbiased are ac- orders an (one-loop) possible in given approximation a under curacy processes com- all fermionic including ap- peting of advantage This important the consideration[14]. has into proach taken is when retardation phonons full of presence the two- in for fermions[13] dimensional analysis (fRG) group renormalization functional system[12]. of type developed electron-phonon originally the apparatus for theoretical our applying h tnadmdlue osuyiieatfermions itinerant study to used model standard The on based is work we which under framework The nefntosadas rvd siae of estimates provide also and functions onse frin emo-oo n boson-boson and fermion-boson -fermion, i reig -aeaddwv pairing d-wave and s-wave ordering, tic tsadtebsn aeBose-Einstein have bosons the and ates U igtefeunydpnec fthe of dependence frequency the ping ytl.W td h hs diagram phase the study We rystals. toa eomlzto ru analysis group renormalization ctional ff lig.W n httnn fthe of tuning that find We filling). eed ndffrn aaeesof parameters different on depends dtebsncflcutoso the of fluctuations bosonic the nd boson-boson , † 87 dmninlotcllattice, optical -dimensional bfrbsn)ta neatthrough interact that bosons) for Rb A92521 CA U bb U eusv neatosand interactions repulsive ) tfie filling fixed at fb ( q )rplieo attrac- or repulsive )) 40 K - 2 type of scattering events (lowest in energy) and opens up is well within experimental capabilities. Additionally, at a large and rich phase space previously inaccessible for half-filling besides the presence of nesting on the Fermi “spinless” fermionic atoms. This type of two-component surface there are van Hove singularities as well that can fermionic system in the presence of a collective mode enhance the interplay and competition among all or- can effectively map and simulate a large variety of corre- ders. Our assumption of BEC for the bosons allows us lated electronic systems in condensed matter physics[5]. to expand the bosonic operators in Eq. (1) according to † The role of isospin in this case is played by different Bq = √N0δq,0 + Bq, where the creation (Bq) or destruc- 40 total angular momentum projection states for the K tion (Bq) of quasiparticles follows from fluctuations (B0) atoms, which can additionally be exploited for Feshbach of the macroscopic number N0 of condensate atoms. Con- resonance type of tuning of the inter-species scattering sequently we can perform the usual Bogoliubov transfor- strength Ufb[15]. mation and linearize the inter-species interaction intro- The Hamiltonian for this system of itinerant fermions ducing the gapless collective mode and bosons on a two-dimensional square lattice has the form ωq = qεq(εq +2nbUbb), (2) H = ξf f † f + ξb B† B b X k k,σ k,σ X q q q where εq = ξq,µb=−4tb is the free-boson dispersion rela- k,σ q tion (chemical potential exactly at the non-interacting 1 † † ground state), and nb = N0/V is the BEC density which + U B BkB ′ Bk′ 2V X bb k+q k −q for all practical purposes can be considered as the bosonic k k′ q , , filling factor on the lattice. 1 † † + Ufb(q)f fk,σB ′ Bk′ This is the fixed point of our theory as far as the sta- V X k+q,σ k −q k,k′,q,σ bility of the bosonic system in the presence of fermions is 1 † † concerned. It has been experimentally shown[16, 17, 18] + U f fk f ′ fk′ , (1) V X ff k+q,↑ ,↑ k −q,↓ ,↓ that the BEC can be stable in the presence of fermionic k,k′,q atoms provided the two-species numbers do not exceed 5 where fk,σ,Bq are the fermion, boson operators respec- 10 atoms in the optical trap in the absence of an optical f,b lattice. The presence of an optical lattice will stabilize tively, ξk = 2tf,b(cos kx + cos ky) µf,b are the corre- sponding dispersion− relations dictated− by the square lat- the system against phase separation or collapse to even higher densities, since both processes rely on the inter- tice symmetry, with tf,b being the overlap integrals and species interaction Ufb dominating over the kinetic en- µf,b the chemical potentials for the fermions and bosons respectively. ergy scales associated with the weakly confined atoms, For the purpose of this study we will focus our at- and three-body recombination processes becoming rel- evant as well. The lattice will introduce experimentally tention at half-filling for the fermions (µf = 0) where the pairing-related phases (s,d-wave) superconductivity tunable kinetic energy scales associated with the hopping (sSC, dSC) can compete with the nesting-related phases of atoms from lattice point to lattice point (tf,b) compet- like charge density wave (CDW) and antiferromagnetism ing with Ufb and consequently stabilizing the system. (SDW), and a rich landscape of phase space becomes Dynamical fluctuations of the BFM around this fixed available for the fermionic system to explore[14]. This point can be modeled in the Matsubara representation. amounts to fermion numbers in the range 103–104 for After the bosonic fields associated with fluctuations of typical optical lattice parameters a = 400nm on 2D op- the BEC are integrated out[11, 12] we arrive at the fully tical harmonic traps of frequency ω =2π 30Hz, which retarded effective fermionic interaction given by ×

2 2 U /Ubb ω U (k , k , k )= U fb k1−k3 , (3) ff 1 2 3 ff 2 2 2 2 − 1+4ξ 2ξ cos(k k )x + cos(k k )y (ω1 ω3) + ω e − 1 − 3 1 − 3
− k1−k3

where ki (iωni , ki), ωni are the fermionic Matsubara nature of the inter-species interaction Ufb[8]. The im- frequencies,≡ and ξ = t /2n U is the healing length portance of retardation is determined by Eq. (2) which p b b bb of the BEC. As we see, fluctuations of the BEC renor- in the longwavelength limit defines the acoustic veloc- malize Uff with an attractive and frequency-dependent ity vs = √2tbUbbnb. For fermionic velocities (vF 2tf ) ≃ part which is not affected by the attractive or repulsive much smaller than vs one recovers the mean-field limit 3

22 22

20 20

18 18

16 16

14 CDW 14 ff ff sSC U 12 U 12 bb bb /U /U 2 fb 2 fb 10 10

U sSC U 8 8

6 6

4 4 dSC 2 2 dSC SDW 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t /t t /t b f b f

FIG. 1: (Color online) Phase diagram for Uff = 0.4tf , Ubb = FIG. 2: (Color online) Phase diagram for Uff = 0.4tf , 0.8tf , and nb = 2.5. Blue circles indicate sSC, red rhombuses Ubb = 0.8tf , and nb = 2.5 in the limit vF ≪ vs where retar- indicate dSC, magenta squares SDW, and green stars CDW dation is not important. In this case, the whole CDW phase type of ordering. Dashed lines are guides to the eye. disappears. Symbols are same as in Fig. (1) and dashed line is guide to the eye.

of these couplings according to[19] for Uff [8]. On the other hand, retardation dominates for ves vF and the competition of phases is enhanced. u(sd)SC = 2(g2 g3), (4) This regime≤ can be achieved experimentally since there − ± uSDW = 2(g4 + g3), (5) are two tunable parameters (tb, Ubb) that can be adjusted uCDW = 2(2g1 + g3 g4), (6) in situ for a given system configuration (nb). The heal- − − ing length ξ plays an important role in the competition of where the signs are chosen so that all orders diverge phases since it defines the length-scale over which bosonic to positive values when they strongly renormalize. In correlations are present[10]. If these correlations, which this two-patch approximation the effective interaction of enter the attractive part of the fermionic interaction, are Eq. (3) reduces to the pairs of short-ranged ξ 1 (in units of the lattice constant) this ≤ 2 2 2 will favor CDW (at half-filling) competing with s-wave Ufb 8tb/ξ g1,3 (ω1,ω2,ω3)=Uff 2 ,(7) type of pairing. If on the contrary ξ > 1, the lattice sym- − U 2 8tb 2 bb (ω ω ) + 2 (1+8ξ ) metry becomes visible and exotic pairing phases enter 1 − 3 ξ 2 the competition[10, 11] along with SDW (at half-filling). Ufb g2,4 (ω1,ω2,ω3)=Uff δω1,ω3 , (8) This is reflected throughout our results as well as we dis- − Ubb cuss below. which are inherently anisotropic (δω1,ω3 is the Kronecker Our fRG analysis for the fermionic system was ap- delta). This inherent anisotropy can enhance d-wave plied at zero temperature for the electron-phonon sys- pairing for example in the regime where g2 < 0,g3 > 0 tem at half-filling[14]. It involves the self-consistent in- with g3 > g2 as Eq. (4) shows. Finally, in order to tegration of fast-modes, determined by the energy cut- account| | for all| frequency| channels correctly one needs to −ℓ off Λ(ℓ) = 4tf e (ℓ is the RG-step in the integra- evaluate the susceptibilities of the different orders and tion process) which renormalizes the fermionic interac- identify the most divergent one[14]. tion until a divergence occurs as the limit ℓ is We have performed our fRG analysis and have inves- approached. At half-filling the presence of van→ ∞Hove tigated various parameter ranges associated with the ex- singularities on the Fermi surface allows us to focus perimentally tunable set of Uff ,Ufb,Ubb,tb,tf ,nb . As { 2 } our attention only on k-space processes around the sin- is reasonably expected, for Ufb/UbbUff < 1 the effective gular points Q1 = (π, 0) + k and Q2 = (0, π) + interaction of Eq. (3) remains repulsive and antiferromag- k with k π, called the two-patch approxima- netic ordering dominates throughout the range of param- | | ≪ 2 tion. As a result, a general fermion-fermion coupling eter space. Interesting physics arises when U /UbbUff fb ≥ Uff (k1, k2, k3) reduces in momentum-space to four non- 1. In Fig. (1) we present the phase diagram our fRG anal- redundant couplings associated with the scattering pro- ysis produces for Uff =0.4tf , Ubb =0.8tf and nb = 2.5 2 cesses of g Uff (Q , Q , Q ), g Uff (Q , Q , Q ), and a range of U /UbbUff > 1, tb/tf values. We notice 1 ≡ 1 2 2 2 ≡ 1 1 1 fb g3 Uff (Q1, Q1, Q2), g4 Uff (Q1, Q2, Q1), where that even at half-filling a large region of the phase dia- we≡ have suppressed the frequency≡ dependence for clar- gram is taken over by pairing related phases. This is the ity. The different orders of interest are defined in terms essential new result of this work and the difference with 4

0.5 fectively amounts to setting ω1 = ω3 for all frequency 0.45 channels in Eqs. (7-8). We have recovered the results at 0.4 half-filling of the previous fRG study[11]. Antiferromag- 2 0.35 netism is expected to take over at the Ufb/UbbUff < 1

0.3 region. As it can be clearly seen by contrasting the two phase diagrams, retardation is very important at tuning

∆ 0.25 the fermionic phases. 0.2 Our analysis has the additional advantage of qualita- 0.15 tively producing the energy gap associated with the dom- −ℓc 0.1 inant orders, ∆ = 4tf e , just by following the critical 0.05 RG-step (ℓc) at which they diverge. We can define

0 0 0.5 1 1.5 2 2.5 3 U 2 2+8ξ2 λ¯ ¯ fb λ = 2 , (9) 2Ubb 1+8ξ FIG. 3: (Color online) Evolution of the gap along tb/tf = 0.6 of Fig. (1) with identical symbol scheme for the different as the average anisotropic coupling strength of the orders. The blue line fitting was according to ∆ = 0.015 + fermion-boson interaction (easily derived from Eqs. (7-8)) 3.326 exp(−(3.101 + λ¯)/λ¯). and investigate the functional form of ∆(λ¯). In Fig. (3) we show the evolution of the gap along the tb/tf = 0.6 direction in Fig. (1), where the fermionic system passes previous studies, where mean-field[10] and vF << vs fRG[11] focused on different dopings for the fermionic through all four phases. The functional form we use for the fitting is ∆ = d + d exp( (d + λ¯)/λ¯) with system. Both were unable to capture at half-filling the 1 2 − 3 rich interplay of orders and their strong dependence on d1 =0.015, d2 =3.326, d3 =3.101. In conclusion we have applied functional renormaliza- tb. As tb is strengthened, the bosonic correlation length ξ tion group method with full retardation for a boson- (which is independent of Ufb) becomes larger allowing for boson-mediated attraction among the fermions to reach fermion mixture system on an optical lattice when the over many lattice sites increasing the Cooper pair sizes fermions are at half-filling and the bosons have Bose- and allowing fermions to avoid on-site pairing and the Einstein condensed. We find a rich phase diagram pro- viding quantitative estimates for the experimental obser- associated energy cost of Uff . For small Ufb values this mechanism is detrimental for CDW and only when the vation of the competing orders. Additionally, we have bosonic correlation length ξ 1/2 can CDW win over produced a functional dependence of the energy gap of ≤ the dominant orders with the average boson-fermion cou- s-wave pairing. For larger values of Ufb charge order can survive over sSC in a region 1/2 ξ < 1. If the range pling strength. of the attractive interaction exceeds≤ the lattice constant We would like to acknowledge fruitful discussions with value then pairing across different sites becomes energet- Karyn Le Hur and Antonio Castro Neto. ically favorable and CDW becomes subleading. At this point we should mention that the ξ 1/2 condition for CDW to appear indicates that the∼ BEC phase has weakened as a whole and long range order has ∗ Electronic address: [email protected] been lost. In free space, the consequence of this is that † Electronic address: [email protected] our linear approximation of coupling fermions to only the [1] M. Greiner et al., Nature 415, 39 (2002) bosonic density fluctuations might not be adequate, there [2] M. Greiner et al., Nature 426, 537 (2003) will be lots of quasiparticle excitations. Nevertheless, we [3] B. Paredes et al., Nature 429, 277 (2004) believe that even if those processes are taken into account [4] Z. Hadzibabic et al., Nature 441, 1118 (2006) the strong influence of the lattice and the presence of [5] M. Lewenstein et al., cond-mat/0606771 [6] H. P. B¨uchler and G. Blatter, Phys. Rev. A 69, 063603 nesting on the Fermi surface will sustain the CDW order (2004); H. P. B¨uchler and G. Blatter, Phys. Rev. Lett. to be experimentally observed. 91, 130404 (2003) As we mentioned in the introduction, all previous [7] M. Lewenstein et al., Phys. Rev. Lett. 92, 050401 (2004) works on this type of system have ignored retardation ef- [8] F. Illuminati and A. Albus, Phys. Rev. Lett 93, 090406 fects. This is an a priori assumption based on comparing (2004) the free-space masses of 40K and 87Rb[10]. When the lat- [9] K. Sengupta, N. Dupuis, and P. Majumdar, tice is turned on, one should be comparing band-masses cond-mat/0603162 [10] D.-W. Wang, M. D. Lukin, and E. Demler, Phys. Rev. A related to tf , tb which can be tuned to any range and con- 72, 051604(R) (2005) sequently vF , vs can be independently set experimentally. [11] L. Mathey et al., Phys. Rev. Lett. 97, 030601 (2006) In Fig. (2) we show the same phase diagram as in Fig. (1) [12] S.-W. Tsai et al., Phys. Rev. B 72, 054531 (2005). without retardation taken into consideration, which ef- [13] R. Shankar, Rev. Mod. Phys. 66, 129 (1994) 5

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