  In this work, we consider excited many-body mean-field states of bosons in a
double-well optical lattice by investigating stationary Bloch solutions to the
non-linear equations of motion. We show that, for any positive interaction
strength, a loop structure emerges at the edge of the band structure whose
existence is entirely due to interactions. This can be contrasted to the case
of a conventional optical (Bravais) lattice where a loop appears only above a
critical repulsive interaction strength. Motivated by the possibility of
realizing such non-linear Bloch states experimentally, we analyze the
collective excitations about these non-linear stationary states and thereby
establish conditions for the system's energetic and dynamical stability. We
find that there are regimes that are dynamically stable and thus apt to be
realized experimentally.
