One-dimensional Bose-Hubbard model with local three-body interactions

We employ the (dynamical) density-matrix renormalization-group technique to investigate the ground-state properties of the Bose-Hubbard model with nearest-neighbor transfer amplitudes $t$ and local two-body and three-body repulsion of strength $U$ and $W$, respectively. We determine the phase boundaries between the Mott-insulating and superfluid phases for the lowest two Mott lobes from the chemical potentials. We calculate the tips of the Mott lobes from the Tomonaga-Luttinger liquid parameter and confirm the positions of the Kosterlitz-Thouless points from the von Neumann entanglement entropy. We find that physical quantities in the second Mott lobe such as the gap and the dynamical structure factor scale almost perfectly in $t/(U+W)$, even close to the Mott transition. Strong-coupling perturbation theory shows that there is no true scaling but deviations from it are quantitatively small in the strong-coupling limit. This observation should remain true in higher dimensions and for not too large attractive three-body interactions.

Figure 1

Ground-state phase diagram of the one-dimensional Bose-Hubbard model with three-body interactions for $W=0$ (left panel), $W=U/2$ (middle panel), and $W=U$ (right panel), showing superfluid (SF) and Mott insulating (MI) regions as a function of the chemical potential $\mu /U$ and the electron transfer amplitude $t/U$. Results are based on DMRG data for lattices with up to $L=128$ sites and open boundary conditions. The position of the Mott tips is obtained from the finite-size extrapolation of the Tomonaga-Luttinger parameter $K_b\left(L\right)$, Eq. (4).

Figure 2

DMRG data for the Tomonaga-Luttinger liquid parameter $K_b\left(L\right)$ (upper panels) and the central charge $c^{*}\left(L\right)$ (lower panels) for the first (left panels) and second (right panels) Mott lobes in the model (1) for $W/U=1$. The position of the maxima of $c^{*}\left(L\right)$ can be extrapolated systematically to the thermodynamic limit, as shown in the insets of the lower panels.

Figure 3

Rescaled single-particle gaps (left panel) and phase diagram (right panel) of the second Mott lobe ($\rho =2$) in the model (1). The data for $W=0,U/2,U$ collapse onto each other. Also shown is the third-order result for the gap ${\Delta }^{\left(3\right)}(U+W,0)$ at $U=W$, Eq. (12).

Figure 4

Dynamical structure factor $S_{\eta }(q,\omega )$ with Lorentz broadening $\eta =2t$ at momenta $q=\pi /2$ (upper panels) and $q=\pi$ (lower panels) of the second Mott lobe in the model (1) for $t/(U+W)=0.05$ (left panels) and $t/(U+W)=0.15$ (right panels), as obtained from the dynamical DMRG technique for $L=32$ sites.

Figure 5

Dynamical structure factor in second-order strong-coupling perturbation theory, $S^{\left(2\right)}(q=\pi ,\omega )$, for $U=W$ and $t/(U+W)=0.15$ (full line), in comparison with the scaling result $S^{(2,\mathrm{sc})}(q=\pi ,\omega )$ (dashed line). The differences are very small.