Quantum phases of attractive bosons on a Bose-Hubbard ladder with three-body constraint

We obtain the complete quantum phase diagram of bosons on a two-leg ladder in the presence of attractive onsite and repulsive interchain nearest-neighbor interactions by imposing the onsite three-body constraint. We find three distinct phases; namely, the atomic superfluid (ASF), dimer superfluid (DSF), and the dimer rung insulator (DRI). In the absence of the interchain nearest-neighbor repulsion, the system exhibits a transition from the ASF to the DSF phase with increasing onsite attraction. However, the presence of the interchain nearest-neighbor repulsion stabilizes a gapped DRI phase, which is flanked by the DSF phase. We also obtain the phase diagram of the system for different values of the interchain nearest-neighbor interaction. By evaluating different order parameters, we obtain the complete phase diagram and the properties of the phase transitions using the self-consistent cluster mean-field theory.

Figure 1

A small section of the ladder model is shown and the cluster contains six sites. Dashed lines represent bonds between decoupled sites, and circles represent lattice sites.

Figure 2

Phase diagram for hardcore bosons in absence of $V$ for different cluster sizes. The cluster sizes are indicated in the legend. The scaled critical point for RMI$(1/2)$-SF transition is represented by a black dot.

Figure 3

Phase diagram for $U<0$ and $V=0.0$. Green (dashed) line and blue (dotted) line indicate first- and second-order transitions, respectively. The first-to-second-order change on the phase boundary is marked by a red circle.

Figure 4

$\rho ,{\rho }_s\mathrm{vs}\mu$ plot for (a) $U=-8.0$, (b) $U=-12.0$, and (c) $U=-15.0$ for $V=0.0$. Solid (red) curves represent average density $\rho$ and circles (green curves) represent average superfluid density ${\rho }_s$ in the cluster.

Figure 5

$U=-11.0,V=0.0$. (a) $\rho ,{\rho }_s\mathrm{vs}\mu$ plot, (b) single-particle tunneling and, (c) pair-tunneling amplitude between sites.

Figure 6

$U=-25.0,V=0.0$. (a) $\rho ,{\rho }_s\mathrm{vs}\mu$ plot, (b) single-particle tunneling and, (c) pair-tunneling amplitude between sites.

Figure 7

$E\left(\varphi \right)-E\left(0\right)$ versus order parameter ($\varphi )$ plot for $U=-9.0,V=0.0$. From top to bottom, $\mu =-6.0,-5.21,-5.114\left({\mu }_c\right),-5.05$, and $-4.95$ for MI(0)-ASF transition across the left-most boundary in Fig. 3.

Figure 8

Plot of $E\left(\varphi \right)-E\left(0\right)$ vs order parameter $\varphi$ for $U=-5.0$. From top to bottom, $\mu =1.21,1.11,1.01\left({\mu }_c\right),0.91$, and 0.81 for ASF-MI(2) transition across the right most boundary in Fig. 3.

Figure 9

$\rho ,{\rho }_s\mathrm{vs}\mu$ plot for (a) $U=-8.0$, (b) $U=-10.5$, and (c) $U=-15.0$ for $V=0.5$. Solid (red) curves represent average density, solid-circle (green) curves represent average superfluid density in the cluster.

Figure 10

Phase diagram for $U<0$ and $V=0.5$ showing different phases. Green (dashed) line and blue (dotted) line indicate first- and second-order transitions, respectively. The first-to-second-order change on the phase boundary is marked by a red dot.

Figure 11

Phase diagram for $U<0$ and $V=1.0$ showing different phases. Green (dashed) line and blue (dotted) line indicate first- and second-order transitions, respectively. The first-to-second-order change on the phase boundary is marked by a red dot. The change in phase boundaries with cluster size is also indicated at $U=-9.0$, $-10.0$, and $-15.0$. For the four-site cluster, the DRI lobe does not extend beyond $U=-9.12$.

Figure 12

$U=-25.0,V=1.0$. (a) $\rho ,{\rho }_s\mathrm{vs}\mu$ plot, (b) single-particle tunneling amplitude between sites, and (c) pair-tunneling amplitude between sites.

Figure 13

$\rho ,{\rho }_s\mathrm{vs}\mu$ plot for HC bosons for (a) $V=0.0$, (b) $V=10.0$, and (c) $V=20.0$.

Figure 14

Phase diagram for hardcore bosons in the presence of interchain nearest-neighbor interaction $V$.

Figure 15

Single-particle tunneling amplitudes for hardcore bosons at (a) $V=0.0$ and (b) $V=20.0$.