  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.
