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

We study a mixture of fermionic and bosonic cold atoms on a two-dimensional optical lattice, where the fermions are prepared in two isospin states and the bosons have Bose-Einstein condensed. Number density fluctuations of the condensate form delocalized bosonic modes which couple to the fermionic atoms similarly to the electron-phonon coupling in crystals. We study the phase diagram for this system at fixed fermion density of one per site. We find that tuning of the lattice parameters and interaction strengths 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. 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.

Figure 1

Phase diagram for $U_{\mathrm{ff}}=0.4t_f$, $U_{\mathrm{bb}}=0.8t_f$, and $n_b=2.5$. Blue circles indicate $s\mathrm{SC}$, red rhombuses indicate $d\mathrm{SC}$, magenta squares SDW, and green stars CDW type of ordering. Dashed lines are guides to the eye.

Figure 2

Phase diagram for $U_{\mathrm{ff}}=0.4t_f$, $U_{\mathrm{bb}}=0.8t_f$, and $n_b=2.5$ in the antiadiabatic limit $v_F\ll v_s$. In this case, the whole CDW phase disappears. Symbols are the same as in Fig. 1 and the dashed line is a guide to the eye.

Figure 3

Evolution of the gap along $t_b/t_f=0.6$ of Fig. 1 with identical symbol scheme for the different orders. The blue line fitting was according to $\Delta =0.015+3.326\exp [-(3.101+\overline{\lambda })/\overline{\lambda }]$.