Staggered ground states in an optical lattice

Nonstandard Bose-Hubbard models can exhibit rich ground-state phase diagrams, even when considering the one-dimensional limit. Using a self-consistent Gutzwiller diagonalization approach, we study the mean-field ground-state properties of a long-range interacting atomic gas in a one-dimensional optical lattice. We first confirm that the inclusion of long-range two-body interactions to the standard Bose-Hubbard model introduces density-wave and supersolid phases. However, the introduction of pair and density-dependent tunneling can result in new phases with two-site periodic density, single-particle transport, and two-body transport order parameters. These staggered phases are potentially a mean-field signature of the known novel twisted superfluids found via a density-matrix renormalization group approach [Phys. Rev. A 94, 011603(R) (2016)]. We also observe other unconventional phases which are characterized by sign staggered order parameters between adjacent lattice sites.

Figure 1

Illustrations of the two-body two-site terms contained in the density-dependent Bose-Hubbard Hamiltonian of Eq. (8). (a) The single-atom tunneling $J$, (b) the on-site two-body interaction $U$, (c) the long-range two-site two-body interaction $V$, (d) the pair tunneling $P$, and (e) the two-body density-dependent tunneling $T$.

Figure 2

Ground-state phase diagram for the standard Bose-Hubbard model ($V/U=T/U=P/U=0$), showing the Mott-insulator–to–superfluid phase transition.

Figure 3

Ground-state phase diagrams with nearest-neighbor interactions present in the system, revealing the presence of inversion symmetry-breaking DW and SS phases. We consider the cases of (a) weak neighbor interactions ($V/U=0.2,T/U=P/U=0$), (b) intermediate neighbor interactions ($V/U=0.4,T/U=P/U=0$), and (c) strong neighbor interactions ($V/U=0.8,T/U=P/U=0$). The bracketed indices next to the DW abbreviation corresponds to the expectation value of the number operators (${\rho }_a, {\rho }_b$) across two sites.

Figure 4

Ground-state phase diagrams with density-dependent tunneling present in the system, considering the cases of (a) local interactions ($V/U=T/U=P/U=0$), (b) weak density-dependent tunneling ($V/U=P/U=0,T/U=-0.005$), and (c) strong density-dependent tunneling ($V/U=P/U=0,T/U=-0.05$). The Mott-insulating phases are observed to be destroyed for increasing $T/U$.

Figure 5

Ground-state phase diagrams with pair tunneling present in the system, considering the cases of (a) local interactions ($V/U=T/U=P/U=0$), (b) weak pair tunneling ($V/U=T/U=0,P/U=0.035$), and (c) strong pair tunneling ($V/U=T/U=0,P/U=0.12$). In a similar manner to the density-dependent tunneling case, the extent of insulating domains is again reduced even for modest $P/U$.

Figure 6

Ground-state phase diagrams with combinations of various interaction processes of interest. We consider the cases of (a) opposite-sign density-dependent tunneling ($V/U=P/U=0,T/U=0.01$), (b) opposite-sign density-dependent tunneling with neighbor interactions ($V/U=0.3,T/U=0.03,P/U=0$), and (c) pair tunneling with neighbor interactions ($V/U=0.3,T/U=0,P/U=0.1$). The emergence of novel superfluid phases such as the OSSF, SP, and PSF can be seen within certain regions. Furthermore, when nearest-neighbor interactions are present, additional long-range, supersolid phases are found (OSSS and PSS).

Figure 7

Phase diagrams and modulation order parameters when $V/U=T/U=0$ and $P/U=0.12$. (a) The detailed staggered phase region of Fig. 5, showing the dominant phases with second-order phase transitions between them. We plot the (b) pair ($\left|\right|{\chi }_1|-|{\chi }_2\left|\right|$) and (c) density ($|{\rho }_1-{\rho }_2|$) modulations, which characterize the three phases within the SP region. The color maps of the pair and density modulation are normalized to 1 from maximum values of 0.7154 and 0.0411, respectively. To highlight the structure of the phases, the color maps are saturated to 0.2.