Resonant superfluidity in the Rabi-coupled spin-dependent Fermi-Hubbard model

We investigate the ground-state phase diagram of the one-dimensional attractive Fermi-Hubbard model with spin-dependent hoppings and an on-site Rabi coupling using the density matrix renormalization group method. In particular, we show that even in the limit of one component being immobile the pair superfluidity can be resonantly enhanced when the Rabi coupling is on the order of the interaction strength just before the system starts to strongly polarize. We derive an effective spin-1/2 XXZ model in order to understand the ground-state properties in the strong attraction limit.

Figure 1

(a) ${\lambda }_{P,0}$ and (b) ${\lambda }_{C,0}$, which measure the SF and CDW QLROs, as a function of $U/t$ and ${\mathrm{\Omega }}_0/U$ for $\delta =0$. (c) and (d) correspond to $U/t_1=30$ with the lines corresponding to $\delta =0$, and the triangles and diamonds corresponding to $\delta =U/2$. (c) shows $n_{\mathrm{pair}}$ (blue line and triangles) and $m=M/N$ (red dashed line and diamonds) whereas the inset corresponds to a zoom-in on the polarized regime. (d) shows the ground-state energy (black line and triangles) and the energy of the fully paired (blue dotted line) and fully polarized (red dashed line) state in the absence of the hopping, where $|E_{\mathrm{pgs}}|=NU/2$ is the unit of energy. All results are for $N=L=100$.

Figure 2

The pair momentum distribution at half filling ($N=L=100$) is plotted as a function of ${\mathrm{\Omega }}_0/U$ ($\delta =0$) for $U=30t_1,t_2=0$.

Figure 3

The ratio $\frac{|t_{\mathrm{eff}}|}{V_{\mathrm{eff}}}$ as a function of $\delta$ and ${\mathrm{\Omega }}_0$ for $t_2=0$. The white space corresponds to $\sqrt{{{\mathrm{\Omega }}_0^2+{\delta }^2}^{\phantom{0}}}>U$ where the perturbation is no longer applicable.

Figure 4

Schematic phase diagram as a function of the number of particles per site $n$ and $\tilde{\mathrm{\Omega }}$ for $t_2=0$ and fixed large attractive interaction $U$ when $\tilde{\mathrm{\Omega }}$ is not too close to zero. For $\tilde{\mathrm{\Omega }}<{\tilde{\mathrm{\Omega }}}_c$ at half filling the system displays CDW long-range order (thick blue line) with a transition to a polarized band insulator (thick orange line) taking place at a critical value. For $n<1$ and $\tilde{\mathrm{\Omega }}<{\tilde{\mathrm{\Omega }}}_c$ the system corresponds to a Luttinger liquid (LL). The dominant QLRO in the LL depends on the filling $n$, and $\tilde{\mathrm{\Omega }}$ and is separated by the line ${\kappa }_{\rho }=1$. For $n<1$ and $\tilde{\mathrm{\Omega }}>1$, the system becomes mostly polarized, and the CDW QLRO is dominant.

Figure 5

All plots correspond to $t_2=0, U=30t_1$. (a) shows the Luttinger parameter as a function of system size at $\tilde{\mathrm{\Omega }}=0.93$ for $n=1/2$ (green diamonds) and $n=1$ (red triangles). (b), (c), and (d) show $P\left(r\right)$ and $C\left(r\right)$ for $\tilde{\mathrm{\Omega }}=0.93,n=1/2, \tilde{\mathrm{\Omega }}=1.13,n=1/2$ and $\tilde{\mathrm{\Omega }}=0.93,n=1$, respectively. The system sizes are given by $L=48$ (blue triangles), $L=96$ (magenta diamonds), and $L=200$ (black squares).

Figure 6

The ground-state properties are plotted by varying the particle number per site $n=N/L$, where $\delta =0$. (a) shows $S\left(\nu \right)/L$ as a function of $n$, where $U=30t_1$ and ${\mathrm{\Omega }}_0=25t_1$. (b) and (c) show ${\lambda }_{P,0}$ (red full line), ${\lambda }_{C,0}$ (blue dashed line), and ${\kappa }_{\rho }$ (black full line) as a function of the filling ratio $n$ for $U=30t_1, {\mathrm{\Omega }}_0=25t_1$, and $U=5t_1,{\mathrm{\Omega }}_0=2.8t_1$, respectively.