Competing Supersolid and Haldane Insulator Phases in the Extended One-Dimensional Bosonic Hubbard Model

The Haldane insulator is a gapped phase characterized by an exotic nonlocal order parameter. The parameter regimes at which it might exist, and how it competes with alternate types of order, such as supersolid order, are still incompletely understood. Using the stochastic Green function quantum Monte Carlo algorithm and density matrix renormalization group, we study numerically the ground state phase diagram of the one-dimensional bosonic Hubbard model with contact and near neighbor repulsive interactions. We show that, depending on the ratio of the near neighbor to contact interactions, this model exhibits charge density waves, superfluid, supersolid, and the recently identified Haldane insulating phases. We show that the Haldane insulating phase exists only at the tip of the unit-filling charge density wave lobe and that there is a stable supersolid phase over a very wide range of parameters.

Figure 1

Shows several quantities for $\rho =1$ as functions of $t/U$ with the fixed ratio $V/U=0.75$. (a) The CDW order parameter for $L=64$ and ${\rho }_s$ for $L=64$, 100, 150; (b) the parity and string order parameters; (c) the neutral and charge gaps. (a) and (b) were obtained with QMC calculations and (c) with DMRG. The region between the vertical dashed lines is the HI. In the HI, ${\rho }_s\to 0$ very slowly as $L$ increases while the string parameter is essentially constant. Note the difference between the neutral and charge gaps. The gaps are given in units of the hopping $t$.

Figure 2

Same as Fig. 1 but at $\rho =2$. (a) QMC simulations show that in the interval between the two vertical (black) dashed lines there is simultaneous SF and CDW order and, therefore, a supersolid phase (SS). The vertical (red) dot-dashed line is where the $L\to \infty$ extrapolated neutral (${\Delta }_n$) and charge (${\Delta }_c$) gaps vanish in DMRG (c). The CDW-SS transition is between $t/U=0.33$ (QMC) and $t/U=0.355$ (DMRG). The difference between the two values could be due to the difference in the boundary conditions, open for DMRG and periodic for QMC. (c) also shows the $L\to \infty$ extrapolated CDW order parameter, right (red) triangles, and the Fourier transform of $⟨n_i⟩⟨n_j⟩$, left (black) triangles, obtained with DMRG to probe the disappearance of CDW order. Both DMRG and QMC give the SS-SF transition at $t/U\approx 0.425$. Note that, unlike Fig. 1, the charge and neutral gaps (c) are essentially always the same.

Figure 3

The dependence of the momentum distribution, $n_k/L$ and the structure factor, $S(k)$ on the system size. $n_k/L\to 0$ with increasing $L$ while $S(k)$ remains constant indicating long-range CDW order.

Figure 4

Main panel: The phase diagram of the BHM obtained with QMC and DMRG simulations. All results are for a system with $V/U=0.75$. All symbols are from QMC and are for $L=64$ sites, $\beta =128$ (stars are for $L=128$). Black lines near the tips of the CDW lobes are DMRG results for $L=192$. Inset: A zoom of the tip of the $\rho =1$ lobe. To the right of the vertical dashed line ($t/U=0.22$) is the HI phase, to the left is the CDW.