Fermion-mediated long-range interactions of bosons in the one-dimensional Bose-Fermi-Hubbard model

The ground-state phase diagram of mixtures of spin polarized fermions and bosons in a 1D periodic lattice is discussed in the limit of large fermion hopping and half filling of the fermions. Numerical simulations performed with the density matrix renormalization group (DMRG) show in addition to bosonic Mott insulating (MI), superfluid (SF), and charge density-wave phases (CDW) a yet unreported phase with spatial separation of MI and CDW regions. We derive an effective bosonic theory which allows for a complete understanding and quantitative prediction of the bosonic phase diagram. In particular the origin of CDW phase and the MI-CDW phase separation is revealed as an effective fermion-mediated long-range interaction between bosons with alternating sign.

Figure 1

Boundaries of the incompressible MI phases and the CDW phase for half fermion filling ${\varrho }_F=1/2$, $J_F=10$, and $V=1.25$ obtained by (open boundary) DMRG and for small values of $J_B$ by (periodic boundary) ED. For the DMRG data, only those data points are used which do not show pinning to the boundary resembling the infinite system case. One recognizes partial overlap between MI and CDW phase for small values of $J_B$ indicating regions of spatial phase separation (PS) between MI and CDW. The dashed lines are results from the second-order perturbation theory based on the effective bosonic model.

Figure 2

Densities of bosons and fermions corresponding to the different regions in the boson phase diagram Fig. 1 for a lattice of length $L=128$ and open boundary conditions. The boson number $N_B$ is (a) 19, (b) 64, (c) 96, and (d) 101. The plots (a), (b), and (c) are taken at $J_B=0.01$ and (d) at $J_B=0.07$. While (b) displays the gaped CDW, (a) and (c) show the PS phase. (d) is outside of the parameter regime with a CDW.

Figure 3

CDW amplitudes (points) of bosons and fermions for double half filling as function of normalized boson hopping. (Dash-dotted line) Analytic estimate of the fermionic amplitude ${\eta }_F$ from the bosonic data. (Dashed line) Analytic estimate of the bosonic amplitude $\eta$ from the fermionic data. One recognizes a reasonable agreement between the numerical data and the analytic estimate, where it should be kept in mind that the underlying perturbation theory gets better for $\eta \to 0$, i.e., for increasing $J_B$.

Figure 4

Boundaries (shifted by $V/2$) of incompressible MI and CDW phases for half fermion filling ${\varrho }_F=1/2$ at vanishing bosonic hopping $J_B=0$ with varying interaction $V$. Curves are the analytic results and the data points are obtained by DMRG (CDW phase, $L=128$) and ED (MI lobes, periodic boundaries with infinite size scaling). The shaded region depicts the coexistence phase between MI and CDW with ${\mu }_0^{+}>{\mu }_{\frac{1}{2}}^{-}$ and ${\mu }_{\frac{1}{2}}^{+}>{\mu }_1^{-}$.