Superfluidity of fermions with repulsive on-site interaction in an anisotropic optical lattice near a Feshbach resonance

We present a numerical study on ground state properties of a one-dimensional (1D) general Hubbard model (GHM) with particle-assisted tunnelling rates and repulsive on-site interaction (positive-U), which describes fermionic atoms in an anisotropic optical lattice near a wide Feshbach resonance. For our calculation, we utilize the time evolving block decimation (TEBD) algorithm, which is an extension of the density matrix renormalization group and provides a well-controlled method for 1D systems. We show that the positive-U GHM, when hole-doped from half-filling, exhibits a phase with coexistence of quasi-long-range superfluid and charge-density-wave orders. This feature is different from the property of the conventional Hubbard model with positive-U, indicating the particle-assisted tunnelling mechanism in GHM brings in qualitatively new physics. The combination of Feshbach resonance and optical-lattice techniques has opened up possibilities for investigating strongly interacting ultracold atoms under tunable configurations [1]. Ability to control such strongly interacting systems provides an unprecedented opportunity to explore interesting states of matter. Much interesting physics has been predicted for ultracold atom systems with fundamental Hubbard model Hamiltonians. For example, with the Bose–Hubbard model and its derivations, people have studied the transition from superfluid to Mott-insulator [2], the existence of supersolid order [3], etc for ultracold bosons, whereas with the Fermi–Hubbard model, Luther–Emery [4] and FFLO [5] phases are predicted to be observable for ultracold fermions with attractive interaction. In particular, it is well known that 1 Author to whom any correspondence should be addressed. New Journal of Physics 10 (2008) 073007 1367-2630/08/073007+07$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 for the repulsive (positive-U) conventional (Fermi–)Hubbard model the susceptibility for superfluid and charge-density-wave (CDW) orders are suppressed at low temperatures and the leading quasi-long-range (QLR) order is given by a spin density wave (SDW) at any filling fraction [6]. However, in this work, we show that coexistence of QLR superfluid and CDW orders can be observed for fermionic atoms with repulsive on-site interaction in an anisotropic optical lattice near a wide Feshbach resonance. The interactions in this strongly interacting system can be described by a one-dimensional (1D), positive-U, general Hubbard model (GHM) with particle-assisted tunnelling rates [7]. The GHM is an effective one-band Hamiltonian that takes into account the multi-band populations and the off-site atom–molecule couplings in an optical lattice near a wide Feshbach resonance (see the detailed derivation in [7]). It is interesting to note that the GHM with similar particle-assisted tunnelling also arises in different physical contexts, as proposed in [8]. In contrast to the case of the conventional positive-U Hubbard model, we show that the superfluid and CDW emerge as dominant QLR orders over spin orders for the positive-U GHM when the system is significantly hole-doped below halffilling, although at or very close to half-filling, the dominant correlation in GHM is still antiferromagnetic. This feature indicates that the particle-assisted tunnelling in GHM brings in qualitatively new physics. It makes the effective interaction in GHM doping-dependent, showing different behaviours with a possible phase-transition in between. We get our results through numerical calculations based on the time-evolving block-decimation (TEBD) algorithm [9, 10], which, as an extension of the density matrix renormalization group (DMRG) method [11], is a well-controlled approach to deal with 1D systems. We compare our numerical results with some known exact results for the conventional Hubbard model and the remarkably precise agreement shows that the calculation here can make quantitatively reliable predictions. As shown in [7], a generic Hamiltonian for describing strongly interacting two-component fermions in an optical lattice (or superlattice) is given by the following GHM: H = ∑ i [ Uni↑ni↓−μni ] − ∑ 〈i, j〉,σ [ t + δg ( niσ + n jσ ) + δtniσn jσ ] a iσa jσ + h.c., (1) where niσ ≡ a † iσaiσ , ni ≡ ni↑ + ni↓, μ is the chemical potential, 〈i, j〉 denotes the neighboring sites and a iσ is the creation operator for generating a fermion on the site i with the spin index σ . The symbol σ stands for (↓,↑) for σ = (↑,↓). The δg and δt terms in the Hamiltonian represent particle-assisted tunnelling, for which the inter-site tunnelling rate depends on whether there is another atom with opposite spin on these two sites. The particle-assisted tunnelling comes from the multi-band population and the off-site atom–molecule coupling for this strongly interacting system [7]. For atoms near a wide Feshbach resonance with the average filling number 〈ni〉6 2, each lattice site could have four different states, either empty (with state |0〉), or a spin ↑ or ↓ atom (a iσ |0〉), or a dressed molecule (d † i |0〉) which is composed of two atoms with opposite spins. The two atoms in a dressed molecule can distribute over a number of lattice bands due to the strong on-site interaction, with the distribution coefficient fixed by solving the single-site problem. One can then mathematically map the dressed molecule state d i |0〉 to a double-occupation state a i↓a † i↑|0〉 by using the atomic operators a † iσ [7]. After this mapping, the effective Hamiltonian is transformed to the form of equation (1). The GHM in equation (1) reduces to the conventional Hubbard model when the particle-assisted tunnelling coefficients δg and δt approaching zero, as one moves far away from the Feshbach resonance. Near the resonance, δg and δt can be significant as compared with the atomic tunnelling rate t due to the renormalization from the multi-band populations and the direct neighboring coupling [7]. New Journal of Physics 10 (2008) 073007 (http://www.njp.org/) 3 We consider in this work an anisotropic optical lattice for which the potential barriers along the xand y-directions are tuned up to completely suppress tunnelling along those directions. The system becomes a set of independent 1D chains. We thus solve the GHM in 1D through numerical analysis. For this purpose, first we transfer all the fermion operators to the hard-core boson operators through the Jordan–Wigner transformation [12]. In the 1D case, we can get rid of the non-local sign factor, and after the transformation the hard-core boson operators satisfy the same Hamiltonian as equation (1). On each site, we then have two hard-core boson modes which are equivalent to a spin-3/2 system with the local Hilbert space dimension d = 4. We can therefore use the TEBD algorithm for solving this pseudo-spin system [9]. Similar to the DMRG method [11], the TEBD algorithm is based on the assumption that in the 1D case the ground state |9〉 = ∑d i1=1 · · · ∑d in=1 ci1...in |i1 · · · in〉 of the Hamiltonian with short-range interactions can be written into the following matrix product form: