Exact diagonalization of the one-dimensional Bose-Hubbard model with local three-body interactions

In this Brief Report the extended Bose-Hubbard model with local two- and three-body interactions is studied by the exact diagonalization approach. The shapes of the first two insulating lobes are discussed and the values of the critical tunneling for which the insulating phase loses stability for repulsive and attractive three-body interactions are predicted.

Figure 1

The critical value of the amplitude of repulsive four-body interactions, $Q_c$, for which the system becomes stable when attractive three-body interactions are present in the system. The exact value of $Q_c$ depends on the filling (dashed line for $\rho =1$ and solid line for $\rho =2$) and it is obtained from the condition that the energy of $n>3$ particles occupying a chosen lattice site must be larger than the energy of the three-particle state at that site.

Figure 2

The upper ${\mu }_{+}$ and lower ${\mu }_{-}$ limits of the insulating phase as a function of the inverse of the system size, $1/L$, for two densities $\rho =1$ (left) and $\rho =2$ (right) (Hamiltonian parameters are $W=U$ and $J/U=0.2$). The solid lines are linear fits to the numerical data points. Extrapolation to infinite system size $1/L\to 0$ gives the energy gap $\Delta$ of the insulating phase in the limit of infinite systems.

Figure 3

Properties of the system for different values of the three-body interaction strengths: $W=0$ (top), $W/U=-0.5$ (middle), and $W/U=1$ (bottom). On the left is the energy gap $\Delta$ of the first ($\rho =1$) and the second ($\rho =2$) MI lobe as a function of hopping amplitude $J$. On the right the phase diagrams of systems considered are presented. Dotted lines represent values of ${\mu }_{-}$ and ${\mu }_{+}$ for an infinite system obtained by extrapolation from finite system properties. Open circles are markers of the transition points, defined as the points where $\Delta$ is close to zero at the level of significance. In the background of each phase diagram the density plots of the correlation function $\Phi$ are visualized. As was expected for $\rho =1$, the insulating phase does not change significantly, but for $\rho =2$ the size of the MI phase crucially depends on the three-body interaction parameter.

Figure 4

Values of the critical tunneling as a function of the three-body interaction parameter $W$ for $\rho =1$ (left) and $\rho =2$ (right). In the first case, the position of the critical point in the phase diagram remains almost unchanged as was previously predicted [17]. In the second case, the position of the critical point depends strongly on the three-body interactions.

Figure 5

(a) The correlation $\Phi =\langle \mathtt{G}\left|a_{i+1}^{†}{a}_i\right|\mathtt{G}\rangle /N$ as a function of tunneling amplitude $J$ for different Hamiltonian parameters and fillings (system size $L=8$). It is seen that for integer fillings (bold solid lines), in contrast to other noninteger ones (dashed and thick solid lines), the function $\Phi$ decays to zero in the limit $J\to 0$. The transition tunneling $J_c$ predicted by the other method occurs when the $\Phi$ curve for integer filling detaches from the curves for fractional fillings. This behavior is universal for repulsive ($W>0$) as well as for attractive ($W<0$) three-body interactions. Note that for legibility a nonlinear scaling has been used. (b) The decay of the energy gap $\Delta$ in the neighborhood of the critical point for repulsive (top) and attractive (bottom) three-body interactions. In chosen variables the data points fit the linear predictions of the Kosterlitz-Thouless phase transition.