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Pairing and Density-Wave Phases in Boson-Fermion Mixtures at Fixed

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Pairing and Density-Wave Phases in Boson-Fermion Mixtures at Fixed Pairing and density-wave phases in Boson-Fermion mixtures at fixed filling F. D. Klironomos∗ and S.-W. Tsai† Physics Department, University of California, Riverside, CA 92521 (Dated: February 3, 2008) We study a mixture of fermionic and bosonic cold atoms on a two-dimensional optical lattice, where the fermions are prepared in two hyperfine (isospin) states and the bosons have Bose-Einstein condensed (BEC). The coupling between the fermionic atoms and the bosonic fluctuations of the BEC has similarities with the electron-phonon coupling in crystals. We study the phase diagram for this system at fixed fermion density of one per site (half-filling). We find that tuning of the lattice parameters and interaction strengths (for fermion-fermion, fermion-boson and boson-boson interactions) drives the system to undergo antiferromagnetic ordering, s-wave and d-wave pairing superconductivity or a charge density wave phase. We use functional renormalization group analysis where retardation effects are fully taken into account by keeping the frequency dependence of the interaction vertices and self-energies. We calculate response functions and also provide estimates of the energy gap associated with the dominant order, and how it depends on different parameters of the problem. PACS numbers: 03.75.Mn,05.10.Cc,71.10.Fd,71.10.Hf Recent advances in manipulating ultracold atoms in applying our theoretical apparatus originally developed optical lattices have produced remarkable results span- for the electron-phonon type of system[12]. ning a variety of modern condensed matter physics phe- The framework under which we work is based on nomena such as the realization of the Mott-insulator functional renormalization group (fRG) analysis for two- transition[1], fermionic superfluidity[2], the realization dimensional fermions[13] in the presence of phonons when of the Tonks-Girardeau gas[3], and the Berezinskii– full retardation is taken into consideration[14]. This ap- Kosterlitz-Thouless transition[4], just to name a few in- proach has the important advantage of including all com- dicative ones. The experimental control over a wide peting fermionic processes under a given (one-loop) ac- range of parameters associated with these type of sys- curacy approximation in an unbiased way, so that all tems renders them a favorite physics playground both possible orders are studied on an equal basis, contrary to for experimentalists and theorists alike. In particular, mean field theory for example which always presupposes a boson-fermion mixture (BFM) system can provide a a manifested order. The only necessary ingredient that fascinating testing ground for a large section of theoreti- is required for this study to work is the bosonic system cal physics[5] furthering our understanding of condensed to be in the Bose-Einstein condensation phase (BEC) matter phenomena. which leads to an electron-phonon type of fermion-boson Previous work has explored part of the vast and rich interaction (linear to first order in the bosonic field)[11]. phase space associated with this two-species system by This allows us to integrate the bosons out and build an employing mean-field or variational type of approaches effective fermionic theory, and consequently apply the and allowing for fluctuating occupation numbers for both fRG analysis to investigate all possible orders of inter- fermions and bosons[6, 7, 8, 9, 10]. As expected, these ap- est the fermions can undergo, while taking retardation proaches provide a good general picture of the rich phase- effects fully into account. What is interesting to realize space but are unable to capture the delicate interplay is that our study can be generalized for any type of a and competition of correlation effects in a quantitative two-component itinerant fermionic system on a square manner which can be a useful reference for experimen- lattice in the presence of a gapless collective mode that talists. A more recent work, based on our methodology of renormalizes attractively the fermionic interaction. All analysis[11], was the first approach to study this type of of our results will remain valid within that context. system beyond mean-field limitations but (as in all pre- The standard model used to study itinerant fermions vious studies) was applied only for the regime vF vs, and bosons on a lattice is the Hubbard model. For a arXiv:cond-mat/0702660v1 [cond-mat.supr-con] 27 Feb 2007 where retardation effects between the fermionic dynam-≪ typical BFM system consisting of neutral atoms (40K 87 ics dictated by the Fermi velocity vF and the bosonic for fermions and Rb for bosons) that interact through dynamics dictated by the sound velocity vs are not im- short-range van der Waals forces a good approximation portant. It is very important to investigate the physics consists only of hardcore (onsite) intra-species (fermion- beyond this regime since vF vs is experimentally ac- fermion Uff , boson-boson Ubb) repulsive interactions and ≥ cessible and in fact, retardation can play a crucial role inter-species (fermion-boson Ufb(q)) repulsive or attrac- in the interplay of orders. The above considerations have tive interactions. Additionally, the presence of a two- prompted us to focus our attention on this type of system component isospin quantum number among fermions re- and complement on the physical understanding of it by laxes Pauli’s exclusion principle limitations for s-wave 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].
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