Quantum degenerate Bose-Fermi mixtures on one-dimensional optical lattices

We study numerically and analytically the phases of a mixture of ultracold bosons and spin-polarized fermions in a one-dimensional lattice. In particular, along a symmetry plane in the parameter space, we obtain the exact boundary of the boson-demixing transition from the Bethe ansatz solution of the standard Hubbard model. For a region of parameter space, we prove the existence of boson-fermion mixed phase at all densities. This phase is a two-component Luttinger liquid for weak couplings or for incommensurate total density, otherwise it has a charge gap but retains a gapless mode of mixture composition fluctuations. We show that the static density correlations in these two regimes are markedly different.

Figure 1

Ground-state phase diagram of the boson-fermion model (1) with $t_B=t_F=1$. Phases are pure bosons (B-B), pure fermions (F-F), and a uniform mixture (B-F). The filled squares joined by dotted lines to guide the eye represent the QMC results, while the solid lines denote the analytic phase boundaries, (2), (3). The dashed lines separate the commensurate (charge-gapped) and noncommensurate (gapless) sectors of the corresponding phases. In the weak coupling limit, the B-F and B-B phases do not have a charge gap.

Figure 2

The density of bosons and fermions at $t_B=t_F=1$ as a function of the fermionic chemical potential ${\mu }_F$.

Figure 3

Charge stiffness of the fermions $\left({\rho }_s^F\right)$ and the phase stiffness of the bosons $\left({\rho }_s^B\right)$ for the parameters as in Fig. 2.

Figure 4

(a) The static charge susceptibility for the fermions ${\chi }_F\left(q\right)$ and bosons ${\chi }_B\left(q\right)$ with varying interspecies interaction strength $V$. The peak at the Fermi wave vector is almost completely destroyed for $V=4$. The densities of the two species are $n_B\approx 0.17$ and $n_F\approx 0.24$. The open (filled) symbols and dashed (solid) lines represent the bosonic (fermionic) data. (b) The susceptibilities in the strong coupling limit at commensurate filling $n_B\approx 0.67$ and $n_F\approx 0.33$.