Figure 1

Ground-state phase diagrams for the Bose-Fermi Hubbard model of Eq. (1) with equal hoppings, i.e., $t\equiv t_b=t_f$. The emerging phases include bosonic superfluid (b-SF), spin-density wave (SDW), polaron-paired counterflow superfluid (PP-CFSF), and phase separation (PS). The phase diagrams from TLL theory (a) and TEBD (b) are depicted in the $(U_{bf}/U_{bb},t/U_{bb})$ plane, where ${\nu }_b={\nu }_f=0.5$. The purple thick solid line represents the phase boundary between the two-component TLL and the Mott insulator, and the strong-coupling regime is achieved below the dashed line. The purple thin solid lines separate the phases in the Mott-insulating region. The green dotted-dashed line in (a) separates the fermionic QLROs, i.e., polaron pairing (PP) and density wave (DW), in the two-component TLL region. The black dotted line in (b) represents $U_{bf}/U_{bb}=1.32$, corresponding to the ${}^{174}$Yb-${}^{173}$Yb mixture of Ref. 9. Our phase boundary is consistent with the results of Ref. [29], shown as an orange dashed-dotted line. (c) The phase diagram from TEBD is depicted in the $(\Delta \nu ,t/U_{bb})$ plane, with $U_{bf}/U_{bb}=1.4$. We take $L=80$ for the TEBD calculations.Reuse & Permissions

Figure 2

Blue squares, green diamonds, and red circles represent the correlation functions $C_{\mathrm{B}}\left(r\right)$, $|C_{\mathrm{P}}\left(r\right)|$, and $C_{\mathrm{PP}}\left(r\right)$, respectively, for $L=80$, $U_{bf}/U_{bb}=1.4$, and $t/U_{bb}=0.4$ (a) and $0.05$ (b). The plots are on a log-log scale.Reuse & Permissions

Figure 3

The exponents ${\alpha }_{\mathrm{B}}$ (blue squares), ${\alpha }_{\mathrm{P}}$ (green diamonds), ${\alpha }_{\mathrm{PP}}$ (red circles), and ${\alpha }_{\mathrm{SDW}}$ (black triangles). While $t/U_{bb}$ is varied at $U_{bf}/U_{bb}=1.4$ in (a), $U_{bf}/U_{bb}$ is varied at $t/U_{bb}=0.05$ in (b). The red solid and green dashed lines represent ${\alpha }_{\mathrm{PP}}=2/K_p$ and ${\alpha }_{\mathrm{P}}=K_p/2+1/\left(2K_p\right)$, which are derived from TLL theory for the 1D polaron gas of Eq. (6) [39]. The Luttinger parameter is given by $K_p=\pi /[2\pi -2{\cos }^{-1}(1-u)]$ at ${\nu }_p=0.5$ [36].Reuse & Permissions

Figure 4

The static structure factor for the spin density $S_{\mathrm{SD}}\left(k\right)$ for $U_{bf}/U_{bb}=0.4$ (blue solid) and $1.4$ (red dashed). We set ${\nu }_b={\nu }_f=0.5$ and $t/U_{bb}=0.05$ at which the system is deep in the Mott-insulating state.Reuse & Permissions

Figure 5

Ground-state properties in the presence of a parabolic trapping potential, where $N_b=N_f=20$, $\Omega /t=0.008$, and $U_{bf}/U_{bb}=1.4$. (a) Density profiles ${\rho }_j$ for several values of $t/U_{bb}$ are shown. (b) The correlation functions $C_{\mathrm{B}}\left(r\right)$ and $C_{\mathrm{PP}}\left(r\right)$ are plotted on a log-log scale, where $t/U_{bb}=0.05$.Reuse & Permissions