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Multi-band and nonlinear hopping corrections to the 3D Bose-Fermi-Hubbard model

Alexander Mering and Michael FleischhauerFachbereich Physik and research center OPTIMAS, Technische Universitat Kaiserslautern, D-67663 Kaiserslautern, Germany

(Dated: April 13, 2011)

Recent experiments revealed the importance of higher-bandeffects for the Mott insulator (MI) – superfluidtransition (SF) of ultracold bosonic atoms or mixtures of bosons and fermions in deep optical lattices [Bestetal., PRL102, 030408 (2009); Willet al., Nature465, 197 (2010)]. In the present work, we derive an effectivelowest-band Hamiltonian in 3D that generalizes the standard Bose-Fermi Hubbard model taking these effects aswell as nonlinear corrections of the tunneling amplitudes mediated by interspecies interactions into account. Itis shown that a correct description of the lattice states in terms of the bare-lattice Wannier functions rather thanapproximations such as harmonic oscillator states is essential. In contrast to self-consistent approaches based oneffective Wannier functions our approach captures the observed reduction of the superfluid phase for repulsiveinterspecies interactions.

PACS numbers: 03.75.Lm, 03.75.Mn, 37.10.Jk, 67.85.Pq

I. INTRODUCTION

Ultracold atoms in optical lattices provide unique andhighly controlable realizations of various many-body Hamil-tonians [1–6]. Theoretical descriptions of these systems inthe case of deep lattice potentials usually employ lowest-bandmodels only [1, 7]. However, it was found recently that for lat-tice bosons with strong interaction contributions to the Hamil-tonian beyond the single-band approximation with nearest-neighbor hopping and local two-particle interactions needtobe taken into account [8]. E.g., using the method ofquan-tum phase diffusion, the value of the two-body interactionU for bosons in a deep optical lattice was measured directlyand found to deviate from the prediction of the tight-bindingmodel derived in [1]. These experiments also revealed thepresence of additional local three- and four-body interactionsnot accounted for in the single-band Bose-Hubbard Hamil-tonian. A perturbative derivation of these terms based onharmonic-oscillator approximations was given by Johnsonetal. [9].

In the case of boson-fermion mixtures, the situation is moreinvolved. The first experiments on mixtures with attractiveinterspecies interaction [10, 11] displayed a decrease of thebosonic superfluidity in the presence of fermions. This initi-ated a controversial discussion about the nature of the effect.Explanations ranged from localization effects of bosons in-duced by fermions [11, 12] to heating due to the admixture[10, 13]. Numerical results also predicted the opposite be-havior, i.e., the enhancement of bosonic superfluidity due tofermions [14] with a more detailed discussion in [15]. The sit-uation remained unclear until a systematic experimental studyof the dependence of the shift in the bosonic SF – MI transi-tion on the boson-fermion interaction [16] and the subsequentobservation of higher-order interactions in the mixture. Thisshows, that again higher-order band effects need to be takeninto account.

The influence of higher Bloch bands in the Bose-Fermimixture can be described by two different approaches: Inthe first approach one assumes that the single-particle Wan-nier functions are altered due to the modification of the latticepotential for one species by the interspecies interaction with

the other [17], which is then calculated in a self-consistentmanner. The agreement of these results to experimentallyobserved shifts of the SF-MI transition is very good for thecase of attractive boson-fermion interaction (see [16]). Themethod fails however for repulsive interactions where exper-iments showed contrary to intuition again a reduction of su-perfluidity [16]. Besides this shortcoming, the self-consistentpotential approach has a conceptual weakness as it can onlybe applied close to the Mott-insulating phase. The secondapproach to include higher bands is an elimination schemeleading to an effective single-band Hamiltonian similar tothepure bosonic case [9, 18, 19]. This approach, although techni-cally more involved, is more satisfactory from a fundamentalpoint of view. It did not result in quantitatively satisfactorypredictions so far, however. We will show here that this isbecause (i) an important non-linear correction to the hoppingmediated by the inter-species interaction and present alreadyin absence of higher-band corrections has been missed out and(ii) harmonic oscillator approximations to the Wannier func-tions which have been used before, lead to gross errors whenconsidering higher band effects.

We here present an adiabatic elimination scheme for Bose-Fermi mixtures obtained independently from [9, 18, 19], re-sulting in an effective first-band BFH-Hamiltonian [20]. Incontrast to [9] and [18, 19] we use correct Wannier functions,which will be shown to be essential. Furthermore we find thatalready within the lowest Bloch band the inter-species interac-tion leads to important nonlinear corrections to the tunnelingmatrix elements of bosons and fermions. For a fixed numberof fermions per site, the effective Hamiltonian is equivalentto the Bose-Hubbard model with renormalized parametersUandJ for which expressions are given in a closed form. Thisallows for a direct study of the influence of the boson-fermioninteractions on the bosonic superfluid to Mott-insulator tran-sition within this level of approximation. It is shown that non-linear hopping together with higher-band corrections leadto areduction of the bosonic superfluidity when adding fermionsfor both, attractiveandrepulsive inter-species interactions.

The outline of the present work is as follows. After deriv-ing the general multi-band Hamiltonian of interacting spin-polarized fermions and bosons in a deep lattice in the follow-

2

ing section, we introduce the first important addition to thestandard BFHM in sectionIII , the nonlinear hopping correc-tion. Restricting to leading contributions, we derive an effec-tive single-band Hamiltonian by adiabatic elimination of thehigher bands in sectionIV. Finally, using the resulting gener-alized BFHM, the effect of a varying boson-fermion interac-tion is studied in detail in sectionV.

II. MODEL

In 3D, ultracold Bose-Fermi mixtures in an external poten-tial are described by the continuous Hamiltonian [7]

H =

∫d3r Ψ†

b(r)

[−

~2

2mb∆+ V b(r)

]Ψb(r)

+

∫d3r Ψ†

f (r)

[−

~2

2mf∆+ V f (r)

]Ψf (r)

+gbb2

∫d3r Ψ†

b(r)Ψ†b(r)Ψb(r)Ψb(r)

+gbf2

∫d3r Ψ†

b(r)Ψ†f (r)Ψf (r)Ψb(r),

(1)

where the index b (f) at the field operatorsΨ refers to bosonic(fermionic) quantities andV b(r) [V f (r)] is the external po-tential consisting of possible trapping potentials as wellasthe optical latticeV b

lat(r) = ηb∑

sin2(kαrα) [V flat(r) =

ηf∑

sin2(kαrα)]. The intra- and interspecies interactionconstants are defined as

gbb =4π~2

mbabb, gbf =

4π~2

mRabf , (2)

with mR =mbmf

mb+mfbeing the reduced mass andabb/bf the

intra- and interspeciess-wave scattering length, respectively.Whereas in the standard approach the field operators in (1)

are expanded in terms of Wannier functions for the first bandonly, we here use an expansion to all Bloch bands:

Ψb(r) =∑

ν

∑

j

bν,j wbν(r− j),

Ψf (r) =∑

ν

∑

j

fν,j wfν(r− j).

(3)

The operatorbν,j [fν,j] denotes the annihilation of a boson

(fermion) in theν-th band at sitej andwb/fν (r − j) is the

corresponding Wannier function of theν-th band located atsite j. The vectorν = {νx, νy, νz} denotes the band index.The Wannier functions factorize as

wb/fν

(r) = wb/fνx (x) wb/f

νy (y) wb/fνz (z) (4)

whith the one-dimensional Wannier functionwb/fβ (x).

Using the expansion of the field operator, the full multi-band Bose-Fermi-Hubbard Hamiltonian can be expressed as:

H =∑

ν,µj1,j2

{J j1j2νµ

b†ν,j1

bµ,j2 + J j1j2νµ

f †ν,j1

fµ,j2

}

+1

2

∑

ν,µ,,σj1...j4

{U j1...j4νµσ

b†ν,j1

b†µ,j2

b,j3 bσ,j4

}

+1

2

∑

ν,µ,,σj1...j4

{V j1...j4ν,µ,,σ b

†ν,j1

bµ,j2 f†,j3

fσ,j4

}.

(5)

The generalized hopping amplitudes (still containing local en-ergy contributions)

J j1j2νµ

=

∫d3r wb

ν(r− j1) ×

×

[−

~2

2mb∆+ V b(r)

]wb

µ(r− j2),

(6)

J j1j2νµ

=

∫d3r wf

ν(r− j1) ×

×

[−

~2

2mf∆+ V f (r)

]wf

µ(r− j2),

(7)

and the generalized interaction amplitudes

U j1j2j3j4νµσ

= gbb

∫d3r wb

ν(r− j1) ×

× wbµ(r− j2)w

b(r− j3)w

bσ(r− j4),

(8)

V j1j2j3j4νµσ

= gbf

∫d3r wb

ν(r− j1) ×

× wbµ(r− j2)w

f(r− j3)w

fσ(r− j4),

(9)

are defined as usual. In the following we restrict our modelin such a way, that only the most relevant terms are kept.Note that many of the matrix elements vanish because ofthe symmetry of the Wannier functions [21]. Unless statedotherwise we restrict ourselves to local contributions ininteraction terms, i.e.j1 = · · · = j4 in U j1,...,j4

νµσandV j1,...,j4

νµσ

and in this case we drop the site indices.

With these restrictions, the general multi-band Hamiltoniancan be cast in the following form

H = H1 +∑

ν 6=1

H0ν+∑′

ν,µ,,σ

Hνµσ, (10)

where the first term

H1 = HBFHM + Hnlin (11)

describes the (pure) first-band (1 = {1, 1, 1}) dynamicsconsisting of the standard Bose-Fermi-Hubbard partHBFHM

[7] and nonlinear hopping correctionsHnlin which will bediscussed in the next section. The second termH0

νincorpo-

rates the (free) dynamics within theν-th band andHνµσ

3

describes the coupling between arbitrary bandsν,µ,,σ.The prime in the sum indicates that at least one multi-indexhas to be different from the others. This general form of thefull Hamiltonian serves as the starting point of our study.

III. NONLINEAR HOPPING CORRECTION

Even when virtual transitions to higher bands are disre-garded there are important corrections to the standard BFHMif the boson-fermion interactionV becomes large. The in-terspecies interaction term in (1) gives rize to a correction tothe bosonic (and fermionic) tunneling amplitude proportionalto the occupation number of the corresponding complemen-tary species. These contributions, in the following termedasnonlinear hopping contributions, have been considered before[22, 23], but have been missed out in earlier discussions ofcorrections to the BFHM [18, 19].

To establish notation let us recall first the usual single-bandBFHM

H = −J∑

〈ij〉

b†i bj +U

2

∑

j

nj (nj − 1)

− J∑

〈ij〉

f †i fj +

V

2

∑

j

njmj.

(12)

The amplitudes are determined by

U ≡ U jjjj1111, V ≡ V jjjj

1111, J ≡ −J j+e,j1,1 , J ≡ −J j+e,j

1,1

with e being an unit vector in one of the three lattice direc-tions. Due to the isotropic setup, the choice of the direction isirelevant. From eq. (8) and (9) two types of nonlinear hoppingcorrections arise: From the boson-boson interaction we obtain

Jbnl

∑

〈ij〉

b†i (ni + nj) bj, (13)

whereas the boson-fermion interaction leads to both, bosonicand fermionic hopping corrections:

Jfnl

∑

〈ij〉

b†i (mi + mj) bj

+ Jnl∑

〈ij〉

f †i (ni + nj) fj.

(14)

The corresponding nonlinear hopping amplitudes read

Jbnl ≡ U j+e,j,j,j

1111 , Jfnl ≡ V j+e,j,j,j

1111 , Jnl ≡ V j,j,j+e,j1111 .

Since we are interested in the influence of the fermions tothe bosons we assume in the following the fermions to be ho-mogenously distributed. This assumption also used in [16, 17]proved to be valid in the trap center and gives a consider-able simplification. This amounts to replacing the fermionicnumber-operators by the fermionic filling:mj → m. Further-more, the bosonic density-operators in eqns. (13) and (14) are

replaced by the filling of the Mott-lobe under consideration,nj → n, for simplicity.

Alltogether, this allows us to write a Hamiltonian includingcorrections from the nonlinear hopping contributions. Defin-ing the effective bosonic hopping amplitude as

J [n,m] ≡ J − 2n Jbnl −mJf

nl, (15)

the system is recast in the form of a pure BHM with densitydependend hopping:

Heff = −J [n,m]∑

〈ij〉

a†i aj +U

2

∑

j

nj (nj − 1) . (16)

Analyzing the resulting predictions for the MI-SF transitionas a function of the filling and the interspecies interaction(seefigure3) one recognizes a substantial reduction of bosonic su-perfluidity for increasing interaction on the attractive side anda corresponding enhancement on the repulsive side, showingthe importance of nonlinear hopping terms for the precise de-termination of the MI–SF transition. Compared to the exper-imental results [16], two main points arise. First, althoughpointing into the right direction for attractive interactions, theoverall shift is too small compared to the experimental ob-servation. Second, for repulsive interactions, the transition isshifted to larger lattice depths, in contrast to the experimentalfindings.

IV. EFFECTIVE SINGLE-BAND HAMILTONIAN

In the following we derive an effective single-band Hamil-tonian that takes into account the coupling to higher bands.The derivation is structured in the following way: We use anadiabatic elimination scheme presented in appendixA whichreduces the main task to the calculation of the the secondorder cumulant〈〈T HI(τ + T )HI(τ)〉〉 in the interactionpicture, where the average is taken over the higher bands.The full interaction HamiltonianHI =

∑′

ν,µ,,σHνµσ is

then reduced according to the relevant contributions of thecumulant. Finally, a reduction of the effective bosonicscattering matrix (A5) gives the full effective single-bandBose-Fermi-Hubbard model.

When calculating the cumulant〈〈T HI(τ + T )HI(τ)〉〉 in(A5), the interaction Hamiltonian of the full multi-band Bose-Fermi-Hubbard model can be reduced considerable. Keepingonly terms that lead to non-zero contributions in lowest or-der, it is easy to see that only those terms inHI matter, whereparticles are transfered to higher bands byHI(τ) and downagain byHI(τ + T ). In the following we restrict ourselves toprecisely those contributions and furthermore treat only localcontributions since these are dominant. Three relevant pro-cesses are found:

1. Single particle transitions to a certain bandν

{1,1,1,1} ↔ {ν,1,1,1}

4

FIG. 1. (Color online) Matrix elements for the coupling of the higherBloch bands to the first band via the generalized interaction(9).Bosonic contributions from (8) are equivalent. Bosons are shownas orange circles and fermions in black.Vν1ν1 describes the tran-sition of a boson and a fermion from the first (higher) to the higher(first) band;Vµ1ν1 gives two particles (boson and fermion) whichperform a transition to bandsν andµ. Vµ111 derscribes a fermion-mediated single particle transition of a boson, whereV11µ1 is aboson-mediated transition of a fermion.

These contributions can be understood as density-mediated band transitions, where the matrix elementsUν111, V

ν111, V1ν11 are only non-zero for odd bandsν. [24] Note that from now on, the upper site-indicesare omitted if they are all the same.

2. Double-transition to the same bandν

{1,1,1,1} ↔ {ν,ν,1,1}

In this situations, two particles undergo a transition tothe same band and all bands are incorporated. The ma-trix elements areUνν11 andVνν11.

3. Double-transition to different bandsν,µ

{1,1,1,1} ↔ {ν,µ,1,1}

In this combined process, the two different bands haveto be both either even or odd with matrix elementsUνµ11 andVνµ11

The remaining important contributions to the full multi-band BFHM result from the kinetic energy of the particles.Restricting to the usual nearest neigbour hoppings within agiven Bloch band (ν = µ and|j1 − j2| = 1) and the energyof the particles within a band (ν = µ andj1 = j2), these are

4. the band energies∆bν

and∆fν

5. the intraband nearest-neighbor hopping for bosonsJνand correspondingly for the fermionsJν .

Hopping between sites with|j1 − j2| 6= 1 is omitted since itis unimportant. In appendixB, the different contributions tothe Hamiltonian as well as the hoppings and band energiesare defined in detail. Figure1 gives a sketch of the differentcontributions taken into account. Shown are only processesinvolving fermions.

From the effective bosonic scattering matrix in (A5), theeffective single-band BFHM is derived by applying a Markovapproximation [25]. This amounts to replacing first-band op-erators at timeτ +T by the corresponding operators at timeτwhich is valid since the timescale of the higher-band dynam-ics is much shorter than in the first band because of the largerhopping amplitude [26]. The resulting Hamiltonian is lengthyand shows the full form is given in appendixC.

The effective Hamiltonian (C1) contains non-local interac-tion and long-range tunneling terms. These result from virtualtransitions into higher bands and subsequent tunneling pro-cesses in these bands. As these terms rapidly decrease withincreasing distance|d| between the involved lattice sites, it issufficient to take into account only the leading order contribu-tions, i.e. only local interaction terms (|d| = 0) and only near-est neigbour hopping(|d| = ±1). This leads to the followingextensions compared to the standard single-band BFHM:

Heff =∑

j

{U3

6nj (nj − 1) (nj − 2) +

V3

2mjnj (nj − 1) +

U2

2nj (nj − 1) +

V2

2njmj

}(17)

+∑

j

∆b1nj +

∑

j

∆b1mj −

∑

〈ij〉

b†i J [ni, nj, mi, mj] bj −∑

〈ij〉

f †i J [ni, nj] fj +

∑

〈ij〉

{J (2)

(b†i

)2b2j + J (2) b†i f

†i fjbj

}.

Here some new terms arize, for instance correlated two-particle tunnelingJ (2) and J (2). Most prominent is theappearance of the three-body interactionsU3 and V3.Thebosonic has recently been measured by means of quantumphase diffusion [8]. It should be noted that in the experimentsin [8] also higher order nonlinear interactions were detected.

Since our approach is only second order in the interaction-induced intra-band coupling, these terms cannot be repro-duced however. Beside the new terms, the higher bands lead toa renormalization of the usual single-band BFHM parameters.Whereas the local two-body interaction amplitudesU2 andV2

only depend on the band structure, the hopping amplitudes are

5

altered, leading to density mediated hopping processes. Forthe bosonic ones, the hopping now is of the form

J [ni, nj, mi, mj] = J − Jbnl (nj + ni)

−Jfnl

2(mj + mi) + α ninj

+ β minj + γ nimj + δ mimj (18)

and the density dependence is directly seen. For all parametersoccuring in (17), full expressions can be found in appendixD.

V. INFLUENCE OF FERMIONS ON THE BOSONIC MI–SFTRANSITION

In order to discuss the phase transition of the bosonic sub-system, we make further approximations. Coming from theMott insulator side of the phase transition, the local num-ber of bosons is approximately given by the integer averagefilling, i.e., 〈nj〉 ≈ n. For the fermionic species, we alsoreplace the number operator by the average fermion numbermj → m = 1, assuming a homogeneous filling of fermionsin the lattice. Having an experimental realization with coldatoms in mind, this is a valid assumption in the center of theharmonic trap at least for attractive inter-species interactions.It should be valid however also for slight inter-species repul-sion. This assumption is also supported by the results of [16],where the actual fermionic density did not influence the tran-sition from a Mott-insulator to a superfluid (for medium andlarge filling). It also agrees with the result in [17] which isbased on this assumption, and which shows a good agree-ment to the experimental results. All further contributionsin the Hamiltonian such as the bosonic three-particle interac-tion and two-particle hoppings are neglected in the following.With these approximations, the renormalized Bose-HubbardHamiltonian for then-th Mott lobe with mean fermionic fill-ingm can be written as

Heff = −J [n,m]∑

〈ij〉

b†i bj +U [m]

2

∑

j

nj (nj − 1) (19)

with

J [n,m] = J − 2n Jbnl −m Jf

nl (20)

−∑

ν 6=1

I eb,ν

(Uν111 n+ V

ν111

m

2

)2

U [m] = U2 +m V3 (21)

The final form of the bosonic Hamiltonian will now be usedto discuss the influence of the boson-fermion interaction onthe Mott-insulator to superfluid transition. Following theex-perimental procedure presented in [16], we consider the shiftof the bosonic transition as a function of the boson-fermioninteraction determined by the scattering lengthaBF , with aspecial emphasis on repulsive interaction where no theoreti-cal prediction exists so far.

potential depth ηB [E

Rec]

ratio

U[1

]/J[1

,1]

5 6 7 8 9 10 11 12 13 14 1510

20

30

40

50

60

708090

100nonlinear bosonicnl bosonic + bands

fermionic shift

pure bosonic

nl bosons + bands + fermions

harmonic approximation

MI to SF transition

FIG. 2. (Color online) Ratio of effective interactionU to effectivetunneling rateJ for unity fermion fillingm = 1 and Mott lobe withn = 1 as function of normalized lattice depthηb for the bosons,and for attractive boson-fermion interaction with a scattering lengthabf = −400 a0. The horizontal dotted line gives the critical valuefor the MF – SF transition point in the Bose-Hubbard model. Shownare the harmonic oscillator approximation together with different lev-els of corrections as described in the main text based on exact Wan-nier functions. Assumed is a perfect match between fermionic andbosonic Wannier functions,ηF ≡ ηB .

To determine the transition point, we calculate the bosonichopping (20) and interaction amplitude (21) using numeri-cally determined Wannier functions. The knowlegde of thecritical ratio U/J of the MI to SF transition from analyticor numerical results [27–29] allows for the precise localiza-tion of the transition point [30]. This method is displayed infigure 2 where the ratio of the effective interaction strengthU [1] and the effective tunneling rateJ [3, 1] as per eq. (20) to(21) are plotted as a function of the normalized lattice depthηb, which describes the amplitude of the periodic lattice po-tential of the bosonsV b

lat in units of the recoil energy of thebosonsEb

rec = ~2k2/(2mb). As indicated, unity fermion fill-

ingm = 1 is assumed and the bosonic Mott lobe withn = 3 isconsidered. The horizontal dotted line gives the critical valuefor the MI – SF transition [27] and the crossing of this linewith the different curves, which illustrate the relative contri-bution of the various correction terms, determines the poten-tial depth at which the phase transition occurs. The differentlevels of approximation shown in figure2 are

1. harmonic oscillator:plain BHM, harmonic oscillator approximation

2. pure bosonic:plain BHM, proper Wannier functions

3. nonlinear bosonic:BHM extended by nonlinear (bosonic) hopping correc-tion

4. nonlinear bosonic with higher bands:inclusion of all bands withνα ≤ 25; this gives the ref-erence point for the shift of the transition

6

5. nonlinear bosonic and fermionic with higher bands:inclusion of fermions; nonlinear hopping correction andhigher bands (να ≤ 25)

One clearly recognizes a substantial shift of the transitionpoint to lower potential depth in qualitative agreement withthe experiment. It is also apparent that using harmonic os-cillator approximations leads to a large error of the predictedtransition point. This shows that the use of the correct Wan-nier functions is crucial for obtaining reliable predictions.

Figure3 shows the shift of the MI – SF transition point forthe first four lobes as a function of the boson-fermion scat-tering lengthabf . The solid lines include all corrections de-scribed earlier, where the amount of the shift is measured rela-tive to the nonlinear bosonic case including higher bands, i.e.,relative to the real bosonic transition point. Thus the figurecorresponds to the shift of the transition point when fermionswith unity filling are added to the system. For each Mott lobethree curves are shown corresponding to different ratios ofηf/ηb which illustrates the effect of different masses and/ordifferent polarizabilities of the bosonic and fermionic speciesas discussed in AppendixE. The dashed-dotted curves givethe contributions of the (first band) nonlinear hopping correc-tions only (bosons and fermions). One recognizes that for in-creasingly attractive interactions between the species there isan increasing shift of the transition point towards smallerpo-tential depth, corresponding to a reduction of bosonic super-fluidity in the presence of fermions. Interestingly one recog-nizes that for repulsive interspecies interactions, virtual tran-sitions to higher Bloch bands tend to counteract the effect ofthe fermion induced nonlinear tunneling. For larger valuesof n there is again a shift of the MI – SF transition point to-wards smaller lattice depth, i.e. again areductionof bosonicsuperfluidity! The latter effect has both been observed in theexperiments [16], but has not been fully understood so far.In the calculations, the bands are summed up to a maximalmulti-indexνmax = {25, 25, 25}, including altogether 15625bands. For this number of bands, a satisfactory convergenceof the effective amplitudesU andJ is found. Overall, oursecond order approach inlcuding the nonlinear correctionsal-ready provides an intuitive explanation for the behaviour ofthe system in the experiment. This especially holds for the re-pulsive case, where the agreement to the experimental resultsis on a quantitative level.

VI. SUMMARY AND OUTLOOK

In the present paper we studied the influence of nonlineartunneling processes and higher Bloch bands on the dynam-ics of a mixture of bosons and fermions in a deep optical lat-tice in a full 3D setup. Taking into account virtual inter-bandtransitions in lowest non-vanishing order and contributions ofthe originally continuous interaction to tunneling processeswe derived an an effective lowest-band Hamiltonian extend-ing the standard Bose-Fermi Hubbard model. This Hamil-tonian contains interaction-mediated nonlinear corrections tothe tunneling rates, renomalized two-body interactions, and

s−wave scattering length aBF

[a0]

MI −

SF

tran

sitio

n po

int η

B [E

rec]

−500 −400 −300 −200 −100 0 100 200 300 400 500

10

11

12

13

14

15

16

17

n=1 n=2 n=3 n=4 pure nl

ηF=0.7 η

B

ηF=1.3 η

B

FIG. 3. (Color online) Shift of the bosonic Mott-insulator to super-fluid transition as a function of the boson-fermion scattering lengthabf for different Mott lobes (solid lines,n = 1 . . . 4, from bottom totop) in one dimension. The gray-shaded region depicts the influenceof a mismatch of the bosonic and fermionic lattice depth. Thedot-dashed lines give the shifts of the transition solely from the nonlin-ear tunneling corrections. Dashed horizontal lines give the transitionpoints for the pure bosonic system.

effective three-body interaction terms. We showed that an ac-curate determination of the effective model parameter requiresthe use of the correct Wannier functions of the correspond-ing single-particle model. As differences in the tails of thewavefunctions are essential, the use of approximate harmonicoscillator wavefunctions can lead to large errors. The effec-tive model allows for a study of the effect of admixing spin-polarized fermionic atoms to the bosonic superfluid to Mott-insulator transition when changing the boson-fermion inter-action strength. Our model recovers qualitatively all featuresobserved in the experiment. In particular we found that bo-son superfluidity is reduced both for attractiveand repulsiveinter-species interactions. The latter has not been reproducedso far with other methods such as the self-consistent potentialapproach.

It should be noted that our model does not take into accountheating effects and effects such as phase separation due to thepresence of an inhomogeneous trapping potential, which haverecently been shown to significantly affect the MI-SF transi-tion point already in the lowest band [31, 32]. We thus expectthat a complete picture of the experimental observations willrequire a proper inclusion of higher-band effects and nonlin-ear tunneling as derived in the present paper, as well as effectsfrom heating and a trapping potential. Finally it should bementioned that our approach is limited to the second order inintra-band processes. In higher-order perturbation theory ef-fective four-body, five-body, etc. interactions will arise, whichplay however a less and less important role. Nevertheless, weexpect that the higher orders should substantially improvetheresults, especially for repulsive interactions.

7

ACKNOLWEDGEMENT

The authors thank S. Das Sarma, I. Bloch and E. Demler foruseful discussions. The financial support by the DFG throughthe SFB-TR49 is gratefully acknowledged.

APPENDIX

Appendix A: Adiabatic elimination scheme

As long as the interaction energiesU and V as well asthe temperature are small compared to the band gap betweenlowest and first excited Bloch band, the population of higherbands can be neglected. However, as noted before, there arevirtual transitions to higher bands which need to be taken intoaccount. In the following we employ an adiabatic elimina-tion scheme of higher Bloch bands starting from the generalmultiband Hamiltonian (10). This scheme, which is also usedin [33] for the Bose-Fermi-Hubbard model in the ultrafast-fermion limit, is equivalent to degenerate perturbation theory[27, 34] and allows for a proper description of the reducedsystem. For this, the Hamiltonian (10) is split up into a freeand an interaction partH = Hfree + HI with

Hfree = H1 +∑

ν 6=1

H0ν, (A1)

HI =∑′

ν,µ,,σ

Hν,µ,,σ. (A2)

Transforming to the interaction picture, the dynamics of thefree part is incorporated by the time dependent interactionHamiltonianHI(τ) = e−

i~Hfreeτ HI e

i~Hfreeτ . Adiabatic

elimination is carried out for the time evolution operator (scat-tering matrix) of the full system given by

S = T exp

{−i

~

∫ ∞

−∞

dτ HI(τ)

}. (A3)

We now trace out the higher-band degrees of freedom, assum-ing empty higher bands. Using Kubo’s cumulant expansion[35]

〈exp{sX}〉X = exp

{∞∑

m=1

sm

m!〈〈Xm〉〉

}(A4)

up to second order in the interband coupling, the effectivescattering matrix for the lowest band reads

Seff = T exp

{(A5)

+1

2

(−i

~

)2 ∫ ∞

−∞

dτ

∫ ∞

−∞

dT 〈〈T HI(τ + T )HI(τ)〉〉

}.

The first order does not lead to any contributions because ofthe vacuum in the higher bands and due to the nature of theinterband couplings. Obviously the effective bosonic Hamil-tonian is connected to the second order cumulants of operatorsin higher Bloch bands,〈〈A B〉〉 = 〈A B〉 − 〈A〉〈B〉 [35].

Appendix B: Relevant band-coupling processes

As discussed in sectionIV, the different terms to the Hamil-tonian are given by

1. Single particle transitions to a certain bandν

1

2

∑

j

[Uν111 b†1b1b

†νb1 + U1ν11 b†1b1b

†νb1+

+U11ν1 b†1bν b†1b1 + U111ν b†1bν b

†1b1+

+Vν111 b†

νb1f

†1f1 + V1ν11 b†1bν f

†1f1+

+V11ν1 b†1b1f†νf1 + V111ν b†1b1f

†1fν

].

(B1)

2. Double-transition to the same bandν

1

2

∑

j

[Uνν11 b†

νb†νb1b1 + U11νν b†1b

†1bν bν+

+Vν1ν1 b†

νb1f

†νf1 + V1ν1ν b†1bν f

†1 fν

].

(B2)

3. Double-transition to different bandsν,µ

1

2

∑

j

[Uνµ11 b†

νb†µb1b1 + U11νµ b†1b

†1bν bµ+

+Vν1µ1 b†

νb1f

†µf1 + V1ν1µ b†1bν f

†1fµ

].

(B3)

Only local contributions are taken into account and thus thespatial indexj is ommited for the moment. The further intra-band contributions are defined as

4. the band energy

∆xν=

∫d3r wx

ν(r)

[−

~2

2mx∆+ V x(r)

]wx

ν(r), (B4)

5. the intraband nearest-neighbor hopping for bosons

Jν =

∫d3r wb

ν(r − e)

[−

~2

2mb∆+ V b(r)

]wb

ν(r) (B5)

and correspondingly for the fermionsJν .

Appendix C: Full effective first-band BFHM

Under the assumptions made in the main text (i. e., onlylocal contributions, nearest-neighbour hopping, etc.), the fi-nal form of the effective Hamiltonian is found from equa-tions (A5) together with the interband couplings from (B1)to (B3)in Markov approximation. This yields

8

Heff1 = H1 +

∑

ν 6=1

∑

jd

{(V

ν1ν1)2 Id

bf,νν

4b†j+df

†j+dfjbj

+ (Uν111)

2 Idb,ν b†j+dnj+d njbj

+Uν111Vν111I

db,ν

2mj+d b†j+dnjbj

+Vν111Uν111I

db,ν

2b†j+dnj+d mjbj

+(V

ν111)2Idb,ν

4mj+d b†j+dmjbj

+(V11ν1)

2 Idf,ν

4nj+d f †

j+dnjfj

}

+∑

ν,µ6=1

∑

jd

{(Uνµ11

)2

4Idbb,νµ

(b†j+d

)2b2j

+

(Vν1µ1

)2

4Idbf,νµb

†j+df

†j+dbjfj

}.

(C1)

In the Hamiltonian, the time integrals over the bosonic andfermionic correlators are defined as

Idb,ν = −

i

~

∫ ∞

0

dT 〈bν,j+d(τ + T )b†ν,j(τ)〉, (C2)

Idbf,νµ = −

i

~

∫ ∞

0

dT 〈bν,j+d(τ + T )b†ν,j(τ)〉×

× 〈fµ,j+d(τ + T )f †µ,j(τ)〉, (C3)

and correspondinglyIdf,ν andId

bb,νµ. The two-point correla-tion functions of bosons and fermions in theν-th band read

〈bν,j+d(τ + T )b†ν,j(τ)〉ν =

1

L3

∑

k

e−2πik·d

L3 ei~Tǫb,ν

k (C4)

〈fν,j+d(τ + T )f †ν,j(τ)〉ν =

1

L3

∑

k

e−2πik·d

L3 ei~Tǫf,ν

k . (C5)

Carrying out the time integration gives in the thermody-namic limit, which is obtained forL → ∞ by settingξ = k

L

and changing1L3

∑k to

∫∫∫d3ξ yields:

Idx,ν =

1

(2π)3

∫∫∫d3ξ

e−iξ·d

ǫx,ν(ξ), (C6)

Idbx,νµ =

1 + δνµδbx(2π)6

∫· · ·

∫d3ξ d3ξ′

e−iξ·de−iξ′·d

ǫb,ν(ξ) + ǫx,µ(ξ′).

(C7)Here

ǫb,ν(ξ) =∑

α=x,y,z

2Jνα cos(ξα) + ∆bνα (C8)

ǫf,ν(ξ) =∑

α=x,y,z

2Jνα cos(ξα) + ∆fνα (C9)

is the energy of a boson respectively fermion in the higherband andx distinguishes between bosons (x = b) andfermions (x = f ).

Appendix D: Definition of constants in Hamiltonian (17)

As used in Hamiltonian (17), the full expressions of thedifferent parameters are:

Density-mediated fermionic or bosonic hopping:

J [nj, nj+e, mj, mj+e] =

J − Jbnl (nj+e + nj)−

Jfnl

2(mj+e + mj)

−∑

ν 6=1

I eb,ν

{(U

ν111)2nj+enj +

Uν111Vν111

2mj+enj

+Vν111Uν111

2nj+emj +

(Vν111)

2

4mj+emj

}. (D1)

J [nj, nj+e] = J −Jnl2

(nj+e + nj)

−∑

ν 6=1

(V11ν1)2I ef,ν

4nj+enj, (D2)

pair tunneling amplitude:

J (2) =U j+e,j+e,j,j1111

2+∑

ν 6=1

(Uνν11)

2I ebb,νν

2(D3)

+∑

ν,µ6=1ν 6=µ

(Uνµ11

)2I ebb,νµ

4,

J (2) =V j+e,j,j+e,j1111

2+∑

ν 6=1

(Vν1ν1)

2I ebf,νν

4(D4)

+∑

ν,µ6=1ν 6=µ

(Vν1µ1)2

4I ebf,νµ,

9

renormalized two-particle interactions:

U2 = U +∑

ν 6=1

(Uν111)

2 I0b,ν +

∑

ν,µ 6=1

1

4

(Uνµ11

)2I0bb,νµ,

(D5)

V2 = V +∑

ν 6=1

(Vν1ν1)

2I0bf,νν

2

+∑

ν 6=1

{(V

ν111)2 I0

b,ν

2+

(V11ν1)2 I0

f,ν

2

}(D6)

+∑

ν,µ6=1ν 6=µ

(Vν1µ1)2

2I0bf,νµ,

three-body interactions

U3 = 6∑

ν 6=1

(Uν111)2I0b,ν , (D7)

V3 =∑

ν 6=1

{Uν111Vν111I

0b,ν +

(V11ν1)2 I0

f,ν

4

}. (D8)

Appendix E: Lattice effects

The lattice potentials for bosons and fermions are both cre-ated by the same laser field and the only externally control-lable parameter is the intensity of this lattice laser. In or-der to see how the parameters of the effective lattice model,such as tunneling rates and interaction constants depend onthis laser intensity one needs to take into account that thereis always a fixed ratiof between the bosonic and fermonicpotential depths for given atomic species and transitions.Todeterminef we note that the optical lattice is generated by anoff-resonant standing laser field. The potential itself resultsfrom the ac-Stark shift. As shown in [36], it is given by

Vpot(r) =3πc2

2

(ΓD1

ω30,D1

∆D1

+2ΓD2

ω30,D2

∆D2

)I(r) (E1)

in rotating wave approximation for a typical alkali D-line dou-blet, where each line contributes independently if the laser issufficiently far detuned from the atomic transitions. The im-portant parameters are the decay ratesΓD1,2

of the excitedstates,∆D1,2

= ωlaser−ω0,D1,2the detunings of the laser fre-

quencyωlaser from the atomic transition frequenciesω0,D1,2

andI(r) = I0 sin2(kr) the laser intensity.Conveniently, all energies in the system are normalized to

the recoil energy of the bosonic species given byEbrec =

~2k2

2mb.

The wavenumberk depends on the chosen optical lattice.The (normalized) lattice potential for the bosons thus reads

V blat(r) = ηb sin

2(kr). It is useful to rewrite the optical latticepotential for the fermionic atoms with respect to the bosonicoptical lattice asV f

lat(r) = ηf sin2(kr), whereηf = f ηb.From eq. (E1) we find

f =

Γf

D1(

ωf

0,D1

)

3

∆f

D1

+2Γf

D2(

ωf

0,D2

)

3

∆f

D2

ΓbD1

(

ωb0,D1

)

3

∆bD1

+2Γb

D2(

ωb0,D2

)

3

∆bD2

. (E2)

At this point, we specify the experimental system. In theprevious discussions, we analyzed the experiment reportedin[16] and use the parameters given there. A mixture of bosonic87Rb and fermionic40K is cooled and put into an optical lat-tice with σL = 755 nm. For Rubidium and Potassium, thetransition wavelengths and decay rates are given by

σKD1

= 766.5 nm σRbD1

= 795.0 nm

ΓKD1

= 38.7× 106 Hz ΓRbD1

= 36.1× 106 Hz (E3)

σKD2

= 769.9 nm σRbD2

= 780.2 nm

ΓKD2

= 38.2× 106 Hz ΓRbD2

= 38.1× 106 Hz.

Using these values,f in equation (E2) evaluates tof =2.04043, which means, that the fermionic lattice potential,in terms of the bosonic recoil energy is twice as deep as thebosonic one. For the calculation of the Wannier functionsof bosons and fermions one has to take into account how-ever also the different masses of the particles. ExpressingtheSchrodinger equation for the single-particle fermionic wave-functionΦf (r) in terms of the bosonic quantitiesηb andmb,one finds[−

~2

2mb∆+

mf

mbf ηb sin2(kr)

]Φf (r) =

mf

mbE Φf (r).

(E4)One recognizes that the difference between the fermionicWannier functions and the bosonic ones is determined onlyby the factormf

mbf . Since for the experiment in [16]

Efrec

Ebrec

=mb

mf= 2.175 (E5)

the factor f is almost compensated,mf

mbf = 0.93. Thus

the bosonic and fermionic Wannier functions are to a goodapproximation identical with a maximal overlap. Neverthe-less, figure3 also display results including a mismatch ofthe bosonic and fermionic Wannier functions, depicted bythe gray shaded regions. The upper (lower) boundary onthe attractive side and the lower (upper) boundary on therepulsive site corresponds to the results for a mismatch ofmf

mbf = 0.7 (

mf

mbf = 1.3), indicating the importance of a

good control of the mismatch in the precise determination ofthe transition shift.

10

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