Competing order in two-band Bose-Hubbard chains with extended-range interactions

Motivated by the recent progress in realizing and controlling extended Bose-Hubbard systems using excitonic or atomic devices, the present Letter theoretically investigates the case of a two-band Bose-Hubbard chain with nearest-neighbor interactions. Specifically, this study concentrates on the scenario where, due to the interactions, one band supports a density-wave phase, i.e., a correlated insulating phase with spontaneous breaking of translational symmetry in the lattice, while the other band supports superfluid behavior. Using the density matrix renormalization group method, we show that supersolid order can emerge from such a combination, that is, an elusive quantum state that combines crystalline order with long-range phase coherence. Depending on the filling of the bands and the interband interaction strength, the supersolid phase competes with phase-separation, superfluid order, or Mott insulating density-wave order. As a possible setup to observe supersolidity, we propose the combination of a lower band supporting density-wave order and a thermally excited band that supports superfluidity due to weaker lattice confinement.

Figure 1

Five possible states at unit filling with negligibly small hopping. These states appear depending on the competition of interactions. In the case of phase separation (PS), the two subsystems may form $n=2$ Mott states (as shown here) or, if $V>3U$, density-wave patterns with fourfold occupied sites (not shown).

Figure 2

The ground-state phase diagram of an unpolarized system ($L=40$) at unit filling with $U_a=5,V_a=4$ and $U_b=1,V_b=0.5$, with $J_a=J_b=1$ being the unit of energy. With this choice, band A is in the DW regime, whereas band B is SF. As a function of interband interactions $V_{ab}$ and $U_{ab}$, we identify different phases due to the interplay of the inequivalent bands. Specifically, the plots show the DW order parameter (color plot), as well as the decay behavior of the superfluid correlation function (blue or red circles, for exponential or algebraic decay, respectively). We separately consider band A (left panel) and band B (middle panel), as well as the entire system jointly (right panel). White regions (with the sky-blue line as a guide for the eye), distinguished by inspecting particle number profiles, are identified as phase-separated regimes, whereas the colored regimes exhibit DW and SF, but also supersolid (SS) phases with coexisting SF and DW orders.

Figure 3

The dependence of the density-wave order parameter and the superfluid correlation function on the particle population in the SS1 regime $U_{ab}=3.2,V_{ab}=0$. The circle lines show the density-wave order parameter in the lower (red), higher (light-blue), and entire (black) band. The red and blue rectangles (circles) show the algebraic and exponential decay of the superfluid correlation function in the lower (upper) band, respectively.

Figure 4

Schematic representation for the implementation of the extended Bose-Hubbard model. Bands A and B are associated to different layers of the optical lattice. $U_{a/b}$ terms correspond to on-site atomic interactions, and extended forces $V_{a/b},U_{ab},V_{ab}$ are induced by dipolar interactions between sites separated along the horizontal, vertical, and diagonal directions, respectively.