Mott transitions in ternary flavor mixtures of ultracold fermions on optical lattices

At the same time, ultracold atoms offer new degrees of freedom: In contrast to the electronic case, with only two spin states, fermionic atoms have large hyperfine multiplets. Early experiments [6] involving three hyperfine states of 40 K have prompted theoretical investigations of 3-flavor mixtures on optical lattices. These studies have focussed on the case of pair-wise equal attractive interactions which may induce trionic phases and exotic superfluidity with a 3-component order parameter, somewhat analogous to QCD [7]. Within the last year, balanced 3flavor 6 Li mixtures have been trapped and studied across Feshbach resonances [8,9]. Soon, such systems, both with repulsive and attractive interactions, should be realized also in optical lattices.

In this paper, we explore the properties of fermionic 3-flavor mixtures on optical lattices in the case of repulsive interactions. While the many ordering patterns conceivable in such systems will certainly warrant extensive studies at some point, ordering phenomena have escaped experimental detection even in the simpler 2-flavor case so far. We will, thus, concentrate on the Mott physics of homogeneous phases. Note that this regime is particularly challenging since it requires nonperturbative theoretical approaches.

Ternary mixtures of fermions on an optical lattice can be modeled via the Hubbard-type Hamiltonian

Here, ij denotes nearest-neighbor sites; α ∈ {1, 2, 3} labels the fermionic flavors, t parameterizes the hopping, U the on-site Coulomb interaction, and µ is the chemical potential. In Eq. ( 1), we have neglected higher Bloch bands and the confining potential. For fermions of a single species (in the vibrational ground state), t α ≈ t α ′ . Following the literature, we will also assume pairwise equal interactions and study the SU(3) symmetric limit t α ≡ t, U αα ′ ≡ U, µ α ≡ µ+U ; by this definition, particlehole symmetry corresponds to a sign change in µ. This system is treated within dynamical mean-field theory (DMFT) which retains the dynamics of local correlations [10]. The semi-elliptic “Bethe” density of states with bandwidth W = 4t * mimics [11] a simple cubic lattice (with t = t * / √ 6); as usual in DMFT studies, the scale is set by the scaled hopping amplitude t * = √ Zt for coordination number Z. The DMFT impurity problem is solved using the Hirsch-Fye quantum Monte Carlo algorithm (HF-QMC) [12], first in the conventional form, with imaginary-time discretization ∆τ ≤ 0.8/U ; the validity of these results is later verified and unbiased results are obtained using the numerically exact multigrid implementation [13]. Note that already conventional HF-QMC is competitive with continuous-time QMC [14]; the multigrid variant is even more efficient. All QMC based methods share the advantage, compared to numerical renormalization group (NRG) approaches, of being reliable at the experimentally relevant (elevated) temperatures.

Results at low temperature -Theoretical studies of electronic 1-band models are often restricted to half filling (µ = 0). Mott metal-insulator transition are, then, signaled by kinks or jumps in thermodynamic properties (as a function of interaction U and/or temperature T ) or, more directly, by the opening of a gap in the local spectral function A(ω). In the present 3-flavor case, genuine Mott physics can be expected only far away from particle-hole symmetry; consequently, the chemical potential µ is an essential additional parameter. In the following, we will choose a relatively low temperature T = t * /20 and explore the (U , µ) space with particular emphasis on properties that are most accessible in quantum gas experiments and which are related to Mott physics. Temperature effects and spectral functions are to be discussed later. Figure 1 shows the filling n = α nα versus the chemical potential µ for a range of on-site interactions U . Initially, for U = 0, n varies smoothly and rapidly with µ from an empty band (n = 0) at µ/t * ≤ -2 to a full band (n = 3) at µ/t * ≥ 2. With increasing U , the slope generally decreases, but the curves remain smooth until, for U/t * ≥ 6, plateaus develop at integer filling n = 1, n = 2 which signal the onset of localized Mott phases (and correspond to gaps in the spectral function). The pa

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