## Quantum phases of Bose-Fermi mixtures in optical lattices

Optics Express, Vol. 12, Issue 1, pp. 55-68 (2004)

http://dx.doi.org/10.1364/OPEX.12.000055

Acrobat PDF (818 KB)

### Abstract

We analyze the ground-state phase diagram of an ultracold Bose-Fermi mixture placed in an optical lattice. The quantum phases involve pairing of fermions and one or several bosons. Depending on the physical parameters these composites can form a Fermi liquid, a density wave, a superfluid or a domain insulator. We determine by means of a mean-field formalism the phase boundaries for finite tunneling, and analyze the experimental feasibility of these sort of phases.

© 2004 Optical Society of America

## 1. Introduction

13. A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, “Observation of Fermi Pressure in a Gas of Trapped Atoms,” Science **291**, 2570 (2001). [CrossRef] [PubMed]

14. F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. **87**, 080403 (2001). [CrossRef] [PubMed]

15. Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Görlitz, and W. Ketterle, “Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases,” Phys. Rev. Lett. **88**, 160401 (2002). [CrossRef] [PubMed]

16. G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, “Collapse of a Degenerate Fermi Gas,” Science **297**, 2240 (2002). [CrossRef] [PubMed]

17. Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, and W. Ketterle, “Fiftyfold Improvement in the Number of Quantum Degenerate Fermionic Atoms,” Phys. Rev. Lett. **91**, 160401 (2003). [CrossRef] [PubMed]

19. K. Mølmer, “Bose Condensates and Fermi Gases at Zero Temperature,” Phys. Rev. Lett. **80**, 1804–1807 (1998). [CrossRef]

20. M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, “Phonon exchange in dilute Fermi-Bose mixtures: Tailoring the Fermi-Fermi interaction,” Phys. Rev. A **61**, 053601 (2000). [CrossRef]

21. H. Pu, W. Zhang, M. Wilkens, and P. Meystre, “Phonon Spectrum and Dynamical Stability of a Dilute Quantum Degenerate Bose-Fermi Mixture,” Phys. Rev. Lett. **88**, 070408 (2002). [CrossRef] [PubMed]

22. P. Capuzzi and E. S. Hernández, “Zero-sound density oscillations in Fermi-Bose mixtures,” Phys. Rev. A **64**, 043607 (2001). [CrossRef]

23. X.-J. Liu and H. Hu, “Collisionless and hydrodynamic excitations of trapped boson-fermion mixtures,” Phys. Rev. A **67**, 023613 (2003). [CrossRef]

24. A. Albus, S. A. Gardiner, F. Illuminati, and M. Wilkens, “Quantum field theory of dilute homogeneous Bose-Fermi mixtures at zero temperature: General formalism and beyond mean-field corrections,” Phys. Rev. A **65**, 053607 (2002). [CrossRef]

25. R. Roth, “Structure and stability of trapped atomic boson-fermion mixtures,” Phys. Rev. A **66**, 013614 (2002). [CrossRef]

26. L. Viverit and S. Giorgini, “Ground-state properties of a dilute Bose-Fermi mixture,” Phys. Rev. A **66**, 063604 (2002). [CrossRef]

27. K. K. Das, “Bose-Fermi Mixtures in One Dimension,” Phys. Rev. Lett. **90**, 170403 (2003). [CrossRef] [PubMed]

28. M. A. Cazalilla and A. F. Ho, “Instabilities in Binary Mixtures of One-Dimensional Quantum Degenerate Gases,” Phys. Rev. Lett. **91**, 150403 (2003). [CrossRef] [PubMed]

8. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature **415**, 39 (2002). [CrossRef] [PubMed]

29. A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A **68**, 023606 (2003). [CrossRef]

29. A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A **68**, 023606 (2003). [CrossRef]

30. H. P. Büchler and G. Blatter, “Supersolid versus Phase Separation in Atomic Bose-Fermi Mixtures,” Phys. Rev. Lett. **91**, 130404 (2003). [CrossRef] [PubMed]

31. R. Roth and K. Burnett, “Quantum phases of atomic boson-fermion mixtures in optical lattices,” http://xxx.lanl.gov/abs/cond-mat/0310114.

32. A. B. Kuklov and B. V. Svistunov, “Counterflow Superfluidity of Two-Species Ultracold Atoms in a Commensurate Optical Lattice,” Phys. Rev. Lett. **90**, 100401 (2003). [CrossRef] [PubMed]

33. M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, “Atomic Bose-Fermi mixtures in an optical lattice,” http://xxx.lanl.gov/abs/cond-mat/0306180.

34. This phenomenon, related to the appearance of counterflow superfluidity in Ref. [32], may occur also in the absence of the optical lattice,M. Yu. Kagan, D. V. Efremov, and A.V. Klaptsov, “Composite fermions in the Fermi-Bose mixture with attractive interaction between fermions and bosons,” http://xxx.lanl.gov/abs/cond-mat/0209481.

33. M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, “Atomic Bose-Fermi mixtures in an optical lattice,” http://xxx.lanl.gov/abs/cond-mat/0306180.

33. M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, “Atomic Bose-Fermi mixtures in an optical lattice,” http://xxx.lanl.gov/abs/cond-mat/0306180.

## 2. Bose-Fermi Hubbard model

^{7}Li-

^{6}Li or

^{87}Rb-

^{40}K. The lattice potential is practically the same for both species in a

^{7}Li and

^{6}Li mixture, and accidentally very similar for the

^{87}Rb and

^{40}K case (for detunings corresponding to the wavelength 1064 nm of a Nd:Yag laser [37]). Due to the periodicity of this potential, the single-atom states form energy bands. If the temperature is low enough and/or the lattice potential wells are sufficiently deep, the atoms occupy only the lowest energy band. Of course, for fermions this is only possible if their number is strictly smaller than the number of lattice sites (filling factor

*ρ*

_{F}≤1). To describe the system under these conditions, we choose a particularly suitable set of single particle states in the lowest energy band, the so-called Wannier states, which are essentially localized at each lattice site. The system is then described by the so-called tight-binding Bose-Fermi-Hubbard (BFH) model (for a derivation from a microscopic model, see e.g. Ref. [29

29. A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A **68**, 023606 (2003). [CrossRef]

38. A. Auerbach, *Interacting Electrons and Quantum magnetism*, (Springer, New York, 1994). [CrossRef]

*b*

_{j},

*f*

_{j}are the bosonic and fermionic creation-annihilation operators, respectively,

*n*

_{i}=

*b*

_{i},

*m*

_{i}=

*f*

_{i}, and

*µ*is the bosonic chemical potential. The fermionic chemical potential is absent in

*H*

_{BFH}, since the fermion number is fixed. The BFH model describes: i) nearest neighbor boson (fermion) hopping, with an associated negative energy, -

*J*

_{B}(-

*J*

_{F}); in Secs. 3 and 4 we shall assume

*J*

_{F}=

*J*

_{B}=

*J*, while the more general case of different tunneling rates will be analyzed in Secs. 5 and 6; ii) on-site repulsive boson-boson interactions with an associated energy

*V*; iii) on-site boson-fermion interactions with an associated energy

*U*, which is positive (negative) for repulsive (attractive) interactions.

## 3. Effective Hamiltonian for composites fermions

*J*≪

*U*,

*V*. We shall show that the zero-temperature physics of this regime can be well described by an effective Hamiltonian for composites fermions consisting in a fermion and a boson(s) (or bosonic hole(s)). We shall discuss in detail the derivation of this effective Hamiltonian, as well as the possible expected quantum phases.

*J*=0), and in absence of Bose-Fermi interactions (

*U*=0), the bosons are in a MI phase with

*ñ*=[

*µ̃*]+1 bosons per site, where [

*µ̃*] is the integer part of

*µ̃*=

*µ*/

*V*. The fermions can be in any available state, since the energy of the system is independent of their configuration. If

*U*> 0 is sufficiently large,

*U*>

*µ*-(

*ñ*-1)

*V*, the fermions push the bosons out of the sites that they occupy, and localized composite fermions are formed, consisting of one fermion and the corresponding number of missing bosons (bosonic holes) (Fig. 1(a)). Similarly, for attractive Bose-Fermi interactions, if

*U*<

*µ*-

*ñ*

*V*, the fermions attract bosons to the sites they occupy, and localized composite fermions are formed, but in this case consisting of one fermion and one or several bosons (Fig. 1(b)).

*α*-

*µ̄*plane, where

*α*=

*U*/

*V*. For

*µ̄*-[

*µ̄*]+

*s*>

*α*>

*µ̄*-[

*µ̄*]+

*s*-1, a single fermion pairs with

*s*holes to form a composite fermion, annihilated by

*s*<0, -

*s*bosons pair with a fermion to generate a composite fermion annihilated by the operator

*i*the number of fermions,

*m*

_{i}, and the number of bosons,

*n*

_{i}, must fulfill the condition

*n*

_{i}+

*sm*

_{i}=

*ñ*.

*f̃*

_{i}fulfill the following anticommutation relation:

*ñ*=

*n*

_{i}+

*m*

_{i}

*s*, we recover the desired fermionic anticommutation relation {

*f̃*

_{i},

*U*=0,

*n̂*, and as a consequence,

*s*cannot be greater than

*ñ*. On the contrary,

*s*can attain arbitrary negative integer values, i.e., we may have fermion composites of one fermion and many bosons in the case of very strong attractive interactions,

*α*<0, and |

*α*|≫1. In Fig. 2(a) the different regions in the phase diagram are denoted with Roman numbers I, II, III, IV etc, which denote the number of particles (fermion + bosons or bosonic holes) that form the corresponding composite fermion. Additionally, a bar over a Roman number indicates composite fermions with one fermion and bosonic holes.

*J*=0 case the phase diagram presents a complex structure. Therefore one can expect a very rich physics for nonzero hopping. The latter can be investigated on the basis of an effective theory for composite fermions, which can be derived using degenerate perturbation theory (to second order in

*J*) along the lines of the derivation of the

*t*-

*J*model (see, e.g., Ref. [38

38. A. Auerbach, *Interacting Electrons and Quantum magnetism*, (Springer, New York, 1994). [CrossRef]

*𝓟*be the projector onto the degenerated ground-state of

*H*

_{0}. This state has the form discussed above. Let

**=1-

*𝓟*. Applying these operators on the time-independent Schrödinger equation

*Eψ*=

*H*

_{BFH}

*ψ*, and expressing

*ψ*as a function of

*ϕ*=

*𝓟ψ*, we obtain:

*H*

_{1}we obtain:

*E*

_{0}is the ground-state energy. One can proceed to evaluate the effective Hamiltonian in the different regions of the phase diagram of Fig. 2. The terms involving only bosonic or only fermionic operators have a major contribution at second order, and lead to an interaction term between composites. Those terms involving the product of bosonic and fermionic operators, lead to hopping terms of the composites from one site to the nearest neighbor. We must note at this point, that at second order only those contributions coming from the regions

*II*and

*III*and

*J*

_{eff}; ii) nearest neighbor composite fermion-fermion interactions with the associated energy

*K*

_{eff}, which may be repulsive (>0) or attractive (<0). In Eq. (6) we employ the number operator

*m̃*

_{i}=

*f̃*

^{†}̃

_{i}

*̃*

*f̃*

_{i}. This effective model is equivalent to that of spinless interacting fermions (c.f., [36, 39

39. R. Shankar, “Renormalization-group approach to interacting fermions,” Rev. Mod. Phys , **66**, 129 (1994). [CrossRef]

*K*

_{eff}has the universal form

*J*

_{eff}on

*J*,

*V*, and

*U*, presents different forms in different regions of the phase diagram in Fig. 2(a). E.g. for 0<

*µ̃*<1:

*K*

_{eff}is always proportional to

*J*

^{2}/

*V*, the dependence of the hopping constant

*J*

_{eff}is different for different number of particles in the composite. This is clear, since we previously commented that only terms of order

*s*bosonic holes (or -

*s*bosons). Hence, the effective hopping depends as

*J*(

*J*/

*V*)

^{s}. The physics of the effective model is determined by the ratio Δ=

*K*

_{eff}/2

*J*

_{eff}, and by the sign of

*K*

_{eff}. In Fig. 2(a) the subindex

*A*(

*R*) denotes attractive (repulsive) interactions.

## 4. Ground-state of the effective Hamiltonian

*H*

_{eff}for the different composites, our initial problem of finding the ground state of the BFH model is reduced to the analysis of the ground state of the spinless Fermi model (6). In this section we discuss the possible quantum phases that may occur in this model.

## 4.1. Repulsive effective interactions

*K*

_{eff}>0. If the filling factor is close to zero,

*ρ*

_{F}≪1, or one, 1-

*ρ*

_{F}≪1, the ground state of

*H*

_{eff}corresponds to a Fermi liquid (a metal), and is well described in the Bloch representation. In addition, in the cases under consideration, the relevant momenta are small compared to the inverse lattice constant (the size of the Brillouin zone). One can thus take the continuous limit, in which the hopping term in

*H*

_{eff}corresponds to a quadratic dispersion with a positive (negative) effective mass for particles (holes), while the nearest neighbor interactions become

*p*-wave interactions. The lattice is irrelevant in this limit, and the system becomes equivalent to a Fermi gas of spinless fermions (for

*ρ*

_{F}≪1), or holes (for 1-

*ρ*

_{F}≪1). Remarkably, this gas is weakly interacting for every value of

*K*

_{eff}, even when

*K*

_{eff}→∞. In the latter case, the sites which surround an occupied site are excluded from the space available for other fermions. As a result, the scattering length remains finite, being of the order of the lattice spacing. Therefore, 1-

*ρ*

_{F}(

*ρ*

_{F}) acts as the gas parameter for the gas of holes (particles). This picture can be rigorously justified using renormalization group approach [39

39. R. Shankar, “Renormalization-group approach to interacting fermions,” Rev. Mod. Phys , **66**, 129 (1994). [CrossRef]

*ρ*

_{F}→1/2, and for large Δ, where the effects of the interactions between fermions become important. One expects then the appearance of localized phases. A physical insight on the properties of this regime can be obtained by using the so-called Gutzwiller ansatz (GA) [40

40. W. Krauth, M. Caffarel, and J.-P. Bouchard, “Gutzwiller wave function for a model of strongly interacting bosons,” Phys. Rev. B **45**, 3137 (1992). [CrossRef]

41. K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, “Superfluid and insulating phases in an interacting-boson model: mean-field theory and the RPA,” Europhys. Lett. **22**, 257 (1993). [CrossRef]

7. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. **81**, 3108 (1998). [CrossRef]

*K*

_{eff}>0 the GA approach maps

*H*

_{eff}onto the classical antiferromagnetic spin model with spins of length 1,

*S*⃗

_{i}=(sin

*θ*

_{i}cos

*ϕ*

_{i}, sin

*θ*

_{i}sin

*ϕ*

_{i},

*cosθ*

_{i}) [38

38. A. Auerbach, *Interacting Electrons and Quantum magnetism*, (Springer, New York, 1994). [CrossRef]

_{crit}=(1+

*m*

_{z}=2

*ρ*

_{F}-1 is the “magnetization per spin”. For Δ<Δ

_{crit}, the ground state is a canted antiferromagnet [38

*Interacting Electrons and Quantum magnetism*, (Springer, New York, 1994). [CrossRef]

_{crit}, the GA ground state of the classical spin model exhibits modulations of

*m*

_{z}with a periodicity of two lattice constants. Coming back to the composite fermion picture, we predict thus that the ground state for Δ < Δ

_{crit}is a Fermi liquid, while for Δ>Δ

_{crit}it is a density wave. For the special case of half filling,

*ρ*

_{F}=1/2, the ground state is the so-called checkerboard state, with every second site occupied by one composite fermion.

_{crit}accurately for

*ρ*

_{F}close to 1/2. One should however stress that the GA value of Δ

_{crit}is incorrect for filling factors

*ρ*

_{F}close to 0 or 1. In particular, the GA approach predicts that Δ

_{crit}tends gradually to infinity and the density wave phase gradually shrinks as

*ρ*

_{F}→0 or 1, i.e. 1-

## 4.2. Attractive effective interactions

*K*

_{eff}<0. In the spin description this corresponds to ferromagnetic spin couplings. In the GA approach the ground state for 0>Δ≥-1 is ferromagnetic and homogeneous. In this description, fixing the fermion number means fixing the

*z*component of the magnetization

*M*

_{z}=

*N*(2

*ρ*

_{F}-1). When |Δ|≪1, and

*ρ*

_{F}is close to zero (one), i.e. low (high) lattice filling, a very good approach to the ground state is given by a BCS Ansatz [42], in which the composite fermions (holes) of opposite momentum build

*p*-wave Cooper pairs,

*v*

_{k}

*̄*and

*u*

_{k}

*̄*are the coefficients of the corresponding Bogoliubov transformation.

*I*,

*II*and

*µ*<1, with the predicted quantum phases. This completes our analysis of the model in 2D and 3D. As we observe the phase diagram in the strong coupling limit is enormously rich, and contains several novel types of quantum phases involving composite fermions, including localized phases (density wave, domain insulator) and delocalized ones (Fermi liquid and superfluid).

## 4.3. One-dimensional case

*x*-

*y*plane will then be

*J*

_{eff}/2, whereas that in the

*z*direction will be

*K*

_{eff}/4. The

*z*component of the magnetization per spin will be fixed by the total number of fermions,

*m*

_{z}=

*ρ*

_{F}-1/2. The ground state of the XXZ chain with a fixed magnetization is somehow similar to the one in a magnetic field, and it is known exactly from the Bethe Ansatz solution [48, 49

49. J. D. Johnson, S. Krinsky, and B. M. McCoy, “Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the X-Y-Z Hamiltonian,” Phys. Rev. A **8**, 2526 (1973). [CrossRef]

## 5. Mean-field theory

50. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B **40**, 546 (1989). [CrossRef]

51. D. van Oosten, P. van der Straten, and H. T. C. Stoof, “Quantum phases in an optical lattice,” Phys. Rev. A **63**, 053601 (2001). [CrossRef]

*T*→0, and

*J*

_{F}≪

*J*

_{B},

*U*,

*V*. We shall analyze only homogeneous phases (it is possible to prove that within the mean-field limit supersolid phases are energetically unfavorable).

*H*

_{BFH}=

*H*

_{0F}+

*H*

_{1B}, where

*H*

_{1B}=-∑〈

_{ij}〉 (

*J*

_{B}

*b*

_{j}+h.c.) and

*H*

_{0F}=

*H*

_{0}-

*J*

_{F}∑〈

_{ij}〉 (

*f*

_{j}+h.c.) describes the fermions moving in the field of the “frozen” self-interacting bosons. We introduce a complex

*c*-number field

*ψ*

_{i}(

*τ*), 0≤

*τ*≤1/

*T*, and decouple the bosonic hopping term

*H*

_{1}by means of the Hubbard-Stratonovich transformation (see e.g. Ref. [36]). In this way the partition function,

*Z*, corresponding to

*H*

_{BFH}can be written in the form

*Z*

_{0}=Tr(-

*H*

_{0}/

*T*) is the partition function corresponding to

*H*

_{0F}. In Eq. 13, we have introduced the effective action

*S*(

*ψ*) is diagonal in the basis of Bloch states, while the second one can be computed as a cumulant expansion in powers of

*ψ*. To determine the superfluid transition point, only the lowest second order in

*ψ*term is relevant. The corresponding contribution reads

_{ij}(

*τ*

_{1}-

*τ*

_{2})=-〈

*T*

_{τ}[

_{1})

*b*

_{j}(τ

_{2})]〉0 is the bosonic Green function corresponding to the Hamiltonian

*H*

_{0F}, which is diagonal in indices

*i*and

*j*,

*𝓖*

_{i j}(τ)=

*δ*

_{ij}

*𝓖*(τ), due to the absence of bosonic hopping in

*H*

_{0F}.

*T*→0, the leading contribution to

*G*(

*τ*) originates from the lowest energy states of the Hamiltonian

*H*

_{0F}. As previously discussed, in the regime of strong coupling,

*U*,

*V*≫

*J*

_{F}, the ground state corresponds to bosonic Mott-Hubbard insulator with

*ñ*bosons per site coexisting with fermions or composite fermions, depending on the ratio between

*U*and

*V*. The fermions could have various phases that are controlled by an effective interfermionic interaction, but these effects are of the order of

*J*

_{F}≪

*U*,

*V*or smaller and therefore can be neglected. As a result, the ground state can be written as

*E*

_{0}=

*E*

_{0}(

*ñ*,0)(1-

*ρ*

_{F})+

*E*

_{0}(

*ñ*-

*s*,1)

*ρ*

_{F}with

*ρ*

_{F}is the fermionic filling factor,

*ñ*the bosonic occupation number in the Mott-Hubbard state, and s the number of bosons (or bosonic holes) participating in the formation of a composite fermion.

*ψ*

_{i}(

*τ*) is a constant,

*ψ*

_{i}(

*τ*)=

*ψ*, and the lowest order contribution to the effective action is

*N*is the number of sites,

*z*the coordinate number of the lattice (

*z*=2

*d*for the considered

*d*dimensional square lattice), and

*𝓖*(

*ω*=0) is the Fourier transform of the Green function at zero frequency. In the considered approximation

*n*,

*m*)=

*E*

_{0}(

*n*+1,

*m*)-

*E*

_{0}(

*n*,

*m*)=

*V*

_{n}+U

_{m}-

*µ*

_{B}. The first line in the above expression corresponds to the sites with no fermions, while the second one comes from the sites with fermions, on which composite fermions are formed. The effective action can now be written in the form

*S*(

*ψ*)=

*N*

_{r}|

*ψ*|

^{2}, where

*r*>0 the system minimizes the energy by having

*ψ*=0 (normal phase), whereas if

*r*<0 a nonzero

*ψ*(superfluid) is energetically favorable. Therefore, the curve

*r*=0 describes the boundaries between the phase with a superfluid bosonic gas, and the interaction-dominated phases. In Fig. 3 we have depicted the curves

*r*=0 for different values of α, and different regions of the

*µ*-

*J*

_{B}phase space.

## 6. Numerical results

*J*

_{F}and

*J*

_{B}can attain arbitrary independent values. In this section we shall discuss in detail our numerical method.

*J*

_{B},

*J*

_{F},

*U*and

*V*[7

7. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. **81**, 3108 (1998). [CrossRef]

*H*

_{BFH}are known, we introduce a GA for the wavefunction:

*nm*〉 are Fock states with

*n*bosons and

*m*fermions, and

*i*. We have checked that the value of the maximal bosonic occupation,

*N*

_{max}, does not affect the calculations. The coefficients

*f*

_{n,m}must satisfy the normalization condition ∑

_{nm}|

*f*

_{n,m}|

^{2}=1. The number of particles is determined by the chemical potential. In the following, we consider only homogeneous phases, although the calculation can be straightforwardly extended to inhomogeneous phases.

_{(k)}=∑

_{nm}

*n*. The chemical potential

*µ*

_{F}is introduced to control the number of fermions.

*U*=0,

*V*=0 and a relative low lattice potential (large

*J*

_{B}). These initial conditions are assumed due to two main reasons. For |

*U*|>0, Eq. (21) is minimized by the non-physical situation in which the number of atoms of one of the species becomes zero; on the contrary a finite number of particles for both species corresponds to a saddle-point of Eq. (21). To provide a system with both species of atoms, the interaction between bosons and fermions must be turned-off for the initial minimization of the energy. In addition for

*J*

_{B}≫

*V*the system is in an homogeneous superfluid phase. With these constraints we obtain appropriate initial conditions for the real-time simulations described below. Due to homogeneity, the ground-state can be found by minimizing the energy on a single cell, while using periodic boundary conditions. The energy is minimized by changing the values of the coefficients

*f*

_{n,m}using a standard downhill technique. In order to guarantee the normalization of the Gutzwiller coefficients, we assume these coefficients as lying on an 2

*N*

_{max}-dimensional hypersphere of unit radius

*f*

_{n,m}coefficients is obtained by employing the proper minimization

*ϕ*(

*i*)=

*n*.

*f*

_{n,m}coefficients in the phase space, and constitute the basis of what has been called dynamic GA [52

52. D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, “Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,” Phys. Rev. Lett. **89**, 040402 (2002). [CrossRef] [PubMed]

*ψ*

_{0}(

*t*=0)

*ψ*

_{1}(

*t*=0) with, respectively, a number of bosons

*N*

_{b}and

*N*

_{b}+

*δN*, where

*δN*≪

*N*

_{b}. We evolve the parallel trajectories while reducing

*J*

_{B}. The chemical potential can be then approximated as (

*µ*(

*t*)≈〈

*ψ*

_{1}(

*t*)|

*H*(

*t*)|

*ψ*

_{1}(

*t*)〉-〈

*ψ*

_{0}(

*t*)|

*H*(

*t*)|

*ψ*

_{0}(

*t*)〉)/

*δN*. By launching various trajectories, we can explore those regions with an incommensurate total number of bosons plus fermions. Consequently, the trajectories do not enter into the regions of the phase diagram in which commensurate phases are expected. Therefore the expected lobular gaps are opened. As shown in Fig. 4, our numerical and analytical results are in rather good agreement. As expected, our numerical method provides worse results at the tip of the lobes and for those phases which are narrower in the

*µ̃*direction (e.g., phases A in Fig. 4(a) and phases B in Fig. 4(b)), since to recover the boundaries close to the tip of the lobe we must employ very close packed numerical trajectories through the phase space, which close to the tips behave badly.

*J*

_{B}=

*J*

_{F}). As expected, the lobes of the phases with composite fermions (phases B in Fig. 4) shrink due to the larger mobility of the fermions.

## 7. Experimental accessibility

43. W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature **413**, 498 (2001). [CrossRef] [PubMed]

44. F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. **87**, 080403 (2001); [CrossRef] [PubMed]

45. A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, “Realization of Bose-Einstein Condensates in Lower Dimensions,” Phys. Rev. Lett. **87**, 130402 (2001). [CrossRef] [PubMed]

46. M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, “Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,” Phys. Rev. Lett. **87**, 160405 (2001); [CrossRef] [PubMed]

47. S. Burger, F. S. Cataliotti, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, “Quasi-2D Bose-Einstein condensation in an optical lattice,” Europhys. Lett. **57**, 1 (2002). [CrossRef]

*J*≪

*V*,

*U*, are fulfilled for sufficiently strong lattice potentials, as those typically employed in current experiments [8

8. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature **415**, 39 (2002). [CrossRef] [PubMed]

*T*=0 analysis is valid for

*T*much lower than the smallest energy scale in our problem, namely the tunneling rate. This experimental constraint is probably the most difficult to fulfill in current experiments. Additionally, the presence of an inhomogeneous trapping potential leads to the appearance of regions of different phases [7

7. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. **81**, 3108 (1998). [CrossRef]

53. G. G. Batrouni, V. Rousseau, R. T. Scalettar, M. Rigol, A. Muramatsu, P. J. H. Denteneer, and M. Troyer, “Mott Domains of Bosons Confined on Optical Lattices,” Phys. Rev. Lett. **89**, 117203 (2002). [CrossRef] [PubMed]

8. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature **415**, 39 (2002). [CrossRef] [PubMed]

5. S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) **392**, 151 (1998). [CrossRef]

6. S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, “Stable 85Rb Bose-Einstein Condensates with Widely Tunable Interactions,” Phys. Rev. Lett. **85**, 1795 (2000). [CrossRef] [PubMed]

*J*≪

*V*, phases

*I*,

*II*and II are easier to study, since the fermions, or composite fermions, attain effective hopping energies that are not too small, and can compete with the effective interactions

*K*

_{eff}. The predicted phases can be detected by using two already widely employed techniques. First, the removal of the confining potentials, and the subsequent presence or absence of interferences in the time of flight image, would distinguish between phase-coherent and incoherent phases. Second, by ramping-up abruptly the lattice potential, it is possible to freeze the spatial density correlations, which could be later on probed by means of Bragg scattering. The latter should allow to distinguish between homogeneous and modulated phases. An independent Bragg analysis for fermions and bosons should reveal the formation of composite fermions.

## 8. Conclusions

## Acknowledgments

## References and links

1. | B. P. Anderson and M. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays”, Science |

2. | O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices,” Phys. Rev. Lett. |

3. | W. K. Hensinger, H. Häffner, A. Browaeys, N. R. Heckenberg, K. Helmerson, C. McKenzie, G. J. Milburn, W. D. Phillips, S. L. Rolston, H. Rubinsztein-Dunlop, and B. Upcroft, “Dynamical tunnelling of ultracold atoms,” Nature |

4. | F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson Junction Arrays with Bose-Einstein Condensates,” Science |

5. | S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) |

6. | S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, “Stable 85Rb Bose-Einstein Condensates with Widely Tunable Interactions,” Phys. Rev. Lett. |

7. | D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. |

8. | M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature |

9. | M. Girardeau, “Relationship between systems of impenetrable bosons and fermions in one dimension,” J. Math. Phys. |

10. | A. Recati, P. O. Fedichev, W. Zwerger, and P. Zoller, “Spin-Charge Separation in Ultracold Quantum Gases,” Phys. Rev. Lett. |

11. | N. K. Wilkin and J. M. F. Gunn, “Condensation of “Composite Bosons” in a Rotating BEC,” Phys. Rev. Lett. |

12. | B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller, “1/2-Anyons in Small Atomic Bose-Einstein Condensates,” Phys. Rev. Lett. |

13. | A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, “Observation of Fermi Pressure in a Gas of Trapped Atoms,” Science |

14. | F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. |

15. | Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Görlitz, and W. Ketterle, “Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases,” Phys. Rev. Lett. |

16. | G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, “Collapse of a Degenerate Fermi Gas,” Science |

17. | Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, and W. Ketterle, “Fiftyfold Improvement in the Number of Quantum Degenerate Fermionic Atoms,” Phys. Rev. Lett. |

18. | G. V. Shlyapnikov, “Ultracold Fermi gases: Towards BCS,” Proc. XVIII Int. Conf. on Atomic Physics, Eds.:H. R. Sadeghpour, D. E. Pritchard, and E. J. Heller, (World Scientific Publishing, Singapore, 2002). |

19. | K. Mølmer, “Bose Condensates and Fermi Gases at Zero Temperature,” Phys. Rev. Lett. |

20. | M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, “Phonon exchange in dilute Fermi-Bose mixtures: Tailoring the Fermi-Fermi interaction,” Phys. Rev. A |

21. | H. Pu, W. Zhang, M. Wilkens, and P. Meystre, “Phonon Spectrum and Dynamical Stability of a Dilute Quantum Degenerate Bose-Fermi Mixture,” Phys. Rev. Lett. |

22. | P. Capuzzi and E. S. Hernández, “Zero-sound density oscillations in Fermi-Bose mixtures,” Phys. Rev. A |

23. | X.-J. Liu and H. Hu, “Collisionless and hydrodynamic excitations of trapped boson-fermion mixtures,” Phys. Rev. A |

24. | A. Albus, S. A. Gardiner, F. Illuminati, and M. Wilkens, “Quantum field theory of dilute homogeneous Bose-Fermi mixtures at zero temperature: General formalism and beyond mean-field corrections,” Phys. Rev. A |

25. | R. Roth, “Structure and stability of trapped atomic boson-fermion mixtures,” Phys. Rev. A |

26. | L. Viverit and S. Giorgini, “Ground-state properties of a dilute Bose-Fermi mixture,” Phys. Rev. A |

27. | K. K. Das, “Bose-Fermi Mixtures in One Dimension,” Phys. Rev. Lett. |

28. | M. A. Cazalilla and A. F. Ho, “Instabilities in Binary Mixtures of One-Dimensional Quantum Degenerate Gases,” Phys. Rev. Lett. |

29. | A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A |

30. | H. P. Büchler and G. Blatter, “Supersolid versus Phase Separation in Atomic Bose-Fermi Mixtures,” Phys. Rev. Lett. |

31. | R. Roth and K. Burnett, “Quantum phases of atomic boson-fermion mixtures in optical lattices,” http://xxx.lanl.gov/abs/cond-mat/0310114. |

32. | A. B. Kuklov and B. V. Svistunov, “Counterflow Superfluidity of Two-Species Ultracold Atoms in a Commensurate Optical Lattice,” Phys. Rev. Lett. |

33. | M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, “Atomic Bose-Fermi mixtures in an optical lattice,” http://xxx.lanl.gov/abs/cond-mat/0306180. |

34. | This phenomenon, related to the appearance of counterflow superfluidity in Ref. [32], may occur also in the absence of the optical lattice,M. Yu. Kagan, D. V. Efremov, and A.V. Klaptsov, “Composite fermions in the Fermi-Bose mixture with attractive interaction between fermions and bosons,” http://xxx.lanl.gov/abs/cond-mat/0209481. |

35. | H. Fehrmann, M. A. Baranov, B. Damski, M. Lewenstein, and L. Santos, “Mean-field theory of Bose-Fermi mixtures in optical lattices,” http://xxx.lanl.gov/abs/cond-mat/0307635. |

36. | S. Sachdev, |

37. | R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys., bf |

38. | A. Auerbach, |

39. | R. Shankar, “Renormalization-group approach to interacting fermions,” Rev. Mod. Phys , |

40. | W. Krauth, M. Caffarel, and J.-P. Bouchard, “Gutzwiller wave function for a model of strongly interacting bosons,” Phys. Rev. B |

41. | K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, “Superfluid and insulating phases in an interacting-boson model: mean-field theory and the RPA,” Europhys. Lett. |

42. | See e.g.P. G. de Gennes |

43. | W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature |

44. | F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. |

45. | A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, “Realization of Bose-Einstein Condensates in Lower Dimensions,” Phys. Rev. Lett. |

46. | M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, “Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,” Phys. Rev. Lett. |

47. | S. Burger, F. S. Cataliotti, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, “Quasi-2D Bose-Einstein condensation in an optical lattice,” Europhys. Lett. |

48. | H. Bethe, “On the Theory of Metals, I. Eigenvalues and Eigenfunctions of a Linear Chain of Atoms,” Z. Phys. |

49. | J. D. Johnson, S. Krinsky, and B. M. McCoy, “Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the X-Y-Z Hamiltonian,” Phys. Rev. A |

50. | M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B |

51. | D. van Oosten, P. van der Straten, and H. T. C. Stoof, “Quantum phases in an optical lattice,” Phys. Rev. A |

52. | D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, “Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,” Phys. Rev. Lett. |

53. | G. G. Batrouni, V. Rousseau, R. T. Scalettar, M. Rigol, A. Muramatsu, P. J. H. Denteneer, and M. Troyer, “Mott Domains of Bosons Confined on Optical Lattices,” Phys. Rev. Lett. |

**OCIS Codes**

(000.2690) General : General physics

(000.6590) General : Statistical mechanics

(020.0020) Atomic and molecular physics : Atomic and molecular physics

**ToC Category:**

Focus Issue: Cold atomic gases in optical lattices

**History**

Original Manuscript: November 10, 2003

Revised Manuscript: January 5, 2004

Published: January 12, 2004

**Citation**

H. Fehrmann, M. Baranov, M. Lewenstein, and Luis Santos, "Quantum phases of Bose-Fermi mixtures in optical lattices," Opt. Express **12**, 55-68 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-55

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### References

- B. P. Anderson and M. Kasevich, �??Macroscopic Quantum Interference from Atomic Tunnel Arrays,�?? Science 282, 1686 (1998). [CrossRef] [PubMed]
- O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, �??Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices,�?? Phys. Rev. Lett. 87, 140402 (2001). [CrossRef] [PubMed]
- W. K. Hensinger, H. Häffner, A. Browaeys, N. R. Heckenberg, K. Helmerson, C. McKenzie, G. J. Milburn, W.D. Phillips, S. L. Rolston, H. Rubinsztein-Dunlop, and B. Upcroft, �??Dynamical tunnelling of ultracold atoms,�?? Nature 412, 52 (2001); [CrossRef] [PubMed]
- F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, �??Josephson Junction Arrays with Bose-Einstein Condensates,�?? Science 293, 843 (2001). [CrossRef] [PubMed]
- S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn,W. Ketterle, �??Observation of Feshbach resonances in a Bose-Einstein condensate,�?? Nature (London) 392, 151 (1998). [CrossRef]
- S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, �??Stable 85Rb Bose-Einstein Condensates with Widely Tunable Interactions,�?? Phys. Rev. Lett. 85, 1795 (2000). [CrossRef] [PubMed]
- D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, �??Cold Bosonic Atoms in Optical Lattices,�?? Phys. Rev. Lett. 81, 3108 (1998). [CrossRef]
- M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I.Bloch, �??Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,�?? Nature 415, 39 (2002). [CrossRef] [PubMed]
- M. Girardeau, �??Relationship between systems of impenetrable bosons and fermions in one dimension,�?? J. Math. Phys. 1, 516 (1960). [CrossRef]
- A. Recati, P. O. Fedichev, W. Zwerger, and P. Zoller, �??Spin-Charge Separation in Ultracold Quantum Gases,�?? Phys. Rev. Lett. 90, 020401 (2003). [CrossRef] [PubMed]
- N. K. Wilkin and J. M. F. Gunn, �??Condensation of �??Composite Bosons�?? in a Rotating BEC,�?? Phys. Rev. Lett. 84, 6 (2000). [CrossRef] [PubMed]
- B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller, �??1/2-Anyons in Small Atomic Bose-Einstein Condensates,�?? Phys. Rev. Lett. 87, 010402 (2001). [CrossRef] [PubMed]
- A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, �??Observation of Fermi Pressure in a Gas of Trapped Atoms,�?? Science 291, 2570 (2001). [CrossRef] [PubMed]
- F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, �??Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,�?? Phys. Rev. Lett. 87, 080403 (2001). [CrossRef] [PubMed]
- Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M.W. Zwierlein, A. Görlitz, andW. Ketterle, �??Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases,�?? Phys. Rev. Lett. 88, 160401 (2002). [CrossRef] [PubMed]
- G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, �??Collapse of a Degenerate Fermi Gas,�?? Science 297, 2240 (2002). [CrossRef] [PubMed]
- Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M.W. Zwierlein, K. Dieckmann, andW. Ketterle, �??Fiftyfold Improvement in the Number of Quantum Degenerate Fermionic Atoms,�?? Phys. Rev. Lett. 91, 160401 (2003). [CrossRef] [PubMed]
- G. V. Shlyapnikov, �??Ultracold Fermi gases: Towards BCS,�?? Proc. XVIII Int. Conf. on Atomic Physics, Eds.: H. R. Sadeghpour, D. E. Pritchard, and E. J. Heller, (World Scientific Publishing, Singapore, 2002).
- K. Mølmer, �??Bose Condensates and Fermi Gases at Zero Temperature,�?? Phys. Rev. Lett. 80, 1804-1807 (1998). [CrossRef]
- M. J. Bijlsma, B. A. Heringa and H. T. C. Stoof, �??Phonon exchange in dilute Fermi-Bose mixtures: Tailoring the Fermi-Fermi interaction,�?? Phys. Rev. A 61, 053601 (2000). [CrossRef]
- H. Pu, W. Zhang, M. Wilkens, and P. Meystre, �??Phonon Spectrum and Dynamical Stability of a Dilute Quantum Degenerate Bose-Fermi Mixture,�?? Phys. Rev. Lett. 88, 070408 (2002). [CrossRef] [PubMed]
- P. Capuzzi and E. S. Hernández, �??Zero-sound density oscillations in Fermi-Bose mixtures,�?? Phys. Rev. A 64, 043607 (2001). [CrossRef]
- X.-J. Liu, and H. Hu, �??Collisionless and hydrodynamic excitations of trapped boson-fermion mixtures,�?? Phys. Rev. A 67, 023613 (2003) [CrossRef]
- A. Albus, S. A. Gardiner, F. Illuminati, and M. Wilkens, �??Quantum field theory of dilute homogeneous Bose-Fermi mixtures at zero temperature: General formalism and beyond mean-field corrections,�?? Phys. Rev. A 65, 053607 (2002). [CrossRef]
- R. Roth, �??Structure and stability of trapped atomic boson-fermion mixtures,�?? Phys. Rev. A 66, 013614 (2002). [CrossRef]
- L. Viverit and S. Giorgini, �??Ground-state properties of a dilute Bose-Fermi mixture,�?? Phys. Rev. A 66, 063604 (2002). [CrossRef]
- K. K. Das, �??Bose-Fermi Mixtures in One Dimension,�?? Phys. Rev. Lett. 90, 170403 (2003) [CrossRef] [PubMed]
- M. A. Cazalilla and A. F. Ho, �??Instabilities in Binary Mixtures of One-Dimensional Quantum Degenerate Gases,�?? Phys. Rev. Lett. 91, 150403 (2003). [CrossRef] [PubMed]
- A. Albus, F. Illuminati and J. Eisert, �??Mixtures of bosonic and fermionic atoms in optical lattices,�?? Phys. Rev. A 68, 023606 (2003). [CrossRef]
- H. P. Büchler and G. Blatter, �??Supersolid versus Phase Separation in Atomic Bose-Fermi Mixtures,�?? Phys. Rev. Lett. 91, 130404 (2003). [CrossRef] [PubMed]
- R. Roth and K. Burnett, �??Quantum phases of atomic boson-fermion mixtures in optical lattices,�?? <a href=" http://xxx.lanl.gov/abs/cond-mat/031011">http://xxx.lanl.gov/abs/cond-mat/031011</a>
- A. B. Kuklov and B. V. Svistunov, �??Counterflow Superfluidity of Two-Species Ultracold Atoms in a Commensurate Optical Lattice,�?? Phys. Rev. Lett. 90, 100401 (2003). [CrossRef] [PubMed]
- M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, �??Atomic Bose-Fermi mixtures in an optical lattice,�?? <a href="http://xxx.lanl.gov/abs/cond-mat/0306180">http://xxx.lanl.gov/abs/cond-mat/0306180</a>
- This phenomenon, related to the appearance of counterflow superfluidity in Ref. [32], may occur also in the absence of the optical lattice, M. Yu. Kagan, D. V. Efremov, and A.V. Klaptsov, �??Composite fermions in the Fermi-Bose mixture with attractive interaction between fermions and bosons,�?? <a href=" http://xxx.lanl.gov/abs/cond-mat/0209481">http://xxx.lanl.gov/abs/cond-mat/0209481</a>
- H. Fehrmann M. A. Baranov, B. Damski, M. Lewenstein, and L. Santos, �??Mean-field theory of Bose-Fermi mixtures in optical lattices,�?? <a href="http://xxx.lanl.gov/abs/cond-mat/0307635">http://xxx.lanl.gov/abs/cond-mat/0307635</a>
- S. Sachdev, Quantum Phase Transitions, (Cambridge University Press, Cambridge, 1999).
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- R. Shankar, �??Renormalization-group approach to interacting fermions,�?? Rev. Mod. Phys, 66, 129 (1994). [CrossRef]
- W. Krauth, M. Caffarel, and J.-P. Bouchard, �??Gutzwiller wave function for a model of strongly interacting bosons,�?? Phys. Rev. B 45, 3137 (1992). [CrossRef]
- K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, �??Superfluid and insulating phases in an interacting-boson model: mean-field theory and the RPA,�?? Europhys. Lett. 22, 257 (1993). [CrossRef]
- See e.g. P. G. de Gennes, Superconductivity in metals and alloys, W. A. Benjamin (1966).
- W. Hänsel, P. Hommelhoff, T. W. Hänsch, J. Reichel, �??Bose-Einstein condensation on a microelectronic chip,�?? Nature 413, 498 (2001). [CrossRef] [PubMed]
- F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, �??Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,�?? Phys. Rev. Lett. 87, 080403 (2001) [CrossRef] [PubMed]
- A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, �??Realization of Bose-Einstein Condensates in Lower Dimensions,�?? Phys. Rev. Lett. 87, 130402 (2001). [CrossRef] [PubMed]
- M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, �??Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,�?? Phys. Rev. Lett. 87, 160405 (2001) [CrossRef] [PubMed]
- S. Burger, F. S. Cataliotti, C. Fort, P. Maddaloni, F. Minardi and M. Inguscio, �??Quasi-2D Bose-Einstein condensation in an optical lattice,�?? Europhys. Lett. 57, 1 (2002). [CrossRef]
- H. Bethe, �??On the Theory of Metals, I. Eigenvalues and Eigenfunctions of a Linear Chain of Atoms,�?? Z. Phys. 74, 205 (1931)
- J. D. Johnson, S. Krinsky, and B. M. McCoy, �??Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the X-Y-Z Hamiltonian,�?? Phys. Rev. A 8, 2526 (1973). [CrossRef]
- M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, �??Boson localization and the superfluid-insulator transition,�?? Phys. Rev. B 40, 546 (1989). [CrossRef]
- D. van Oosten, P. van der Straten, and H. T. C. Stoof, �??Quantum phases in an optical lattice,�?? Phys. Rev. A 63, 053601 (2001). [CrossRef]
- D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, �??Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,�?? Phys. Rev. Lett. 89, 040402 (2002) [CrossRef] [PubMed]
- G. G. Batrouni, V. Rousseau, R. T. Scalettar, M. Rigol, A. Muramatsu, P. J. H. Denteneer, and M. Troyer, �??Mott Domains of Bosons Confined on Optical Lattices,�?? Phys. Rev. Lett. 89, 117203 (2002). [CrossRef] [PubMed]

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