Collisionless drag for a one-dimensional two-component Bose-Hubbard model

We theoretically investigate the elusive Andreev-Bashkin collisionless drag for a two-component one-dimensional Bose-Hubbard model on a ring. By means of tensor network algorithms, we calculate the superfluid stiffness matrix as a function of intra- and interspecies interactions and of the lattice filling. We then focus on the most promising region close to the so-called pair-superfluid phase, where we observe that the drag can become comparable with the total superfluid density. We elucidate the importance of the drag in determining the long-range behavior of the correlation functions and the spin speed of sound. In this way, we are able to provide an expression for the spin Luttinger parameter $K_S$ in terms of drag and the spin susceptibility. Our results are promising in view of implementing the system by using ultracold Bose mixtures trapped in deep optical lattices, where the size of the sample is of the same order of the number of particles we simulate. Importantly, the mesoscopicity of the system, far from being detrimental, appears to favor a large drag, avoiding the Berezinskii-Kosterlitz-Thouless jump at the transition to the pair-superfluid phase which would reduce the region where a large drag can be observed.

Figure 1

Sketch of the two-component Bose-Hubbard ring with hopping parameters ${\tilde{t}}_{\alpha }=t_{\alpha }{e}^{-i2\pi {\varphi }_{\alpha }/L}$, on-site intraspecies interactions $U_{\alpha }$, and interspecies interaction $U_{AB}$. The two fluxes ${\varphi }_{\alpha }$ pierce the ring and give rise to bosonic currents $j_{\alpha }$ in the ground state.

Figure 2

The superfluid currents $j_A$ and $j_B$ in the presence of ${\varphi }_A$ only for a $L=32$ system. The four sets of curves (from dark to light color) correspond to $U_{AB}/U=0.0,0.25,0.5,0.75$, with $U/t=2$ at half filling. The drag density $n_{AB}^{\left(s\right)}$ is proportional to the slope of the $j_B$ curve in the limit of ${\varphi }_A\to 0$. The inset demonstrates the global momentum conservation because of the constant value of the total current for all values of $U_{AB}$.

Figure 3

Superfluid drag in terms of the total density (main panel) and of the total superfluid density (inset). The red dashed line indicates the theoretical prediction via Bogoliubov approximation [19]. The relevant pairing correlations are responsible for the nonsymmetric trend exhibited by the attractive regime ($U_{AB}<0$).

Figure 4

Normalized drag with respect to the total superfluid density for different system sizes, as a function of $U_{AB}$ for a system with $U=10t$ and $\nu =1$. From bottom to top (from light to dark shades), $L=8,16,32,64,96$, with corresponding bond dimensions $\chi =100,300,600,700,800$. Points are data (with error bars explained in the Supplemental Material [26]); lines are an artistic guide to the eye. The thermodynamic limit should exhibit a saturation to $\left(n_{AB}^{\left(s\right)}\right)/(n_{AA}^{\left(s\right)}+n_{AB}^{\left(s\right)})=0.5$ in the PSF region. In the inset, the same points of the main plot are represented as a function of the inverse of the system's size $L^{-1}$ for different values of $-0.30\le U_{AB}/U\le -0.15$, from top to bottom.

Figure 5

The correlation functions of Eq. (9) [(a) $G_{\alpha }$, (b) $R_D$, and (c) $R_S$] as a function of the conformal distance for a unitary total filling system $\nu =1$, $t=1$, and $U=10$. The orange points concern a regime in which the system is in a PSF phase, while the blue ones concern the 2SF phase, and we represent, with a color gradient from dark to light, different system sizes from $L=8$ to $L=64$. The dashed lines are exponential and algebraic fits, depending on the expected behavior of the functions for $U_{AB}/U=-0.1$, and the solid lines are for $U_{AB}/U=-0.5$.

Figure 6

(a) Luttinger parameter $K_S$ for the spin channel, as obtained from the hydrodynamic relation in Eq. (13) (solid line with error shadow) and from the correlation functions, given by Eq. (9) (points with error bars). Points are reported until the algebraic fit makes sense: the shaded region indicates where deviations become sizable (for more details, see the Supplemental Material [26]). In the PSF, the parameter $K_S$ must go to zero. (b) The behavior of the susceptibility as a function of the interaction, as estimated from various system sizes (color code as in Fig. 4, except for $L=8$ which is omitted).