Anomalous pairing of bosons: Effect of multibody interactions in an optical lattice

A first-order type phase transition between Mott lobes has been reported in Phys. Rev. Lett. 109, 135302 (2012) for a two-dimensional Bose-Hubbard model in the presence of an attractive three-body interaction. We revisit the scenario in systems of ultracold bosons both in one- and two-dimensional lattices using the density matrix renormalization group method and the self-consistent cluster mean-field theory approach, respectively. We show that an unconventional pairing of particles occurs due to the competing repulsive two-body and attractive three-body interactions. This leads to a pair superfluid phase sandwiched between the Mott insulator lobes corresponding to densities density one and three in the strongly interacting regime. This is in contrast to the direct first-order jump as predicted before. Interestingly, the Mott to pair superfluid phase transitions are found to be continuous in nature. We also show that the pair superfluid phase is robust with respect to the ratio between the two- and three-body interaction strengths. In the end, we establish a connection between the Bose-Hubbard model presented in Phys. Rev. Lett. 109, 135302 (2012) with a more general Bose-Hubbard model and analyze the fate of the pair superfluid phase in the presence of an external trapping potential.

Figure 1

(a) Phase diagram for the model given in Eq. (1) for $W/U=-1.5$, obtained using the DMRG method. Inset shows the zoomed in part of the main phase diagram, (b) MI phase at $\rho =1.0$, (c) PSF phase, and (d) MI phase at $\rho =3.0$.

Figure 2

(a) DMRG $\rho -\mu$ plots at $t/U=0.053$ and $W/U=-1.5$ with $L=40$ and 80 sites. The blue curve (with larger steps) corresponds to $L=40$ and the magenta curve (with smaller steps) corresponds to $L=80$ sites. The dashed lines mark the regions of various phases, and (b) and (c) are zoomed in regions depicting the ASF and PSF phases, respectively.

Figure 3

Parity order parameter for the PSF to ASF transition as function of $(t/U)$ using various system sizes ($L$) at $\rho =2$. (Inset) Collapse of all finite size data to one curve close to the phase transition point ${(t/U)}_c\approx 0.045$ according to typical Ising scaling relations with $\beta =1/8$ and $\nu =1$.

Figure 4

Scaling of the fidelity susceptibility ${\chi }_{\text{FS}}/L$ as function of $t/U$. The inset shows the linear divergence of the peak height of the ${\chi }_{\text{FS}}/L$ curve with $L$.

Figure 5

Plot of single particle correlation function ($G$) and pair correlation function ($G^p$) with respect to the distance ($r$) at $t/U=0.053$. The numbers in brackets denote the density at which respective correlations are calculated.

Figure 6

(a) Parity order parameter $O_P^2$ as function of the three-body interaction strength $W/U$ for different system sizes $L=20$ (green triangles), $L=40$ (red dots), and $L=80$ (blue squares) at filling $n=1$ and $t/U=0.03$. The momentum distributions $n{\left(k\right)}_{\text{single}}$ (red solid lines) and $n{\left(k\right)}_{\text{pair}}$ (black dashed lines), for $W/U=-2.0,-1.0$, and $-0.5$ are shown in (b), (c), and (d), respectively.

Figure 7

Phase diagram for the model given in Eq. (1) for $W/U=-1.5$, obtained using the CMFT approach. Solid and dashed lines indicate the first- and second-order phase transition boundaries, respectively. Green circles indicate the tricritical points. The MI(1) and MI(3) lobes are demarcated by the red and blue boundaries. The region between the MI lobes bounded by the dashed green line shows the PSF phase.

Figure 8

(a) $\rho ,{\rho }_s-\mu$ plot at $zt/U=0.05$ of Fig. 7. (b) $C_{1p}$ and $C_{2p}$ showing the clear signature of the PSF phase (see text).

Figure 9

The $\rho -\mu$ plots for different values of $W/U$: (a) $-1.2$, (b) $-1.4$, (c) $-1.6$, and (d) $-1.8$. For all these ratios the value of $zt/U$ is fixed as 0.02.

Figure 10

$E\left(\varphi \right)-E(\varphi =0)$ vs $\varphi$ around the critical chemical potential (${\mu }_c=18.2217$) across the MI-ASF phase boundary for $zt/U=0.14$. The merging of three degenerate minima indicates a first-order transition which is further confirmed by the hysteresis curve (inset) behavior at this point.

Figure 11

$E\left(\varphi \right)-E(\varphi =0)$ vs $\varphi$ around the critical chemical potential (${\mu }_c=12.7828$) across the MI-ASF phase boundary for $zt/U=0.17$. The merging of two degenerate minima into one indicates a second-order transition.

Figure 12

Density and order parameters with second-order tunneling terms for a cut along $\mu /U=0.751$ in Fig. 7. Order parameters vary in a continuous manner across MI(3)-PSF and PSF-ASF phase boundaries indicating that these are second-order phase transitions.

Figure 13

Scaling of the CMFT results for $L=2$, 4, and 6 sites in the thermodynamic limit. The $\mu /U$ values considered are (top to bottom) $0.4896,0.6631$, and 0.7702, respectively. The symbol $\times$ denotes the critical points obtained after extrapolating the 2-, 4-, and 6-site results. Data from QMC are marked by $\square$ on the $y$ axis.

Figure 14

MI(1), PSF, and MI(3) phases in the presence of external confining potential $V_{\text{trap}}=0.002$ and at $zt/U=0.05, \mu /U=0.753125$.