  We show the violation of the entanglement-area law for bosonic systems with
Bose surfaces. For bosonic systems with gapless factorized energy dispersions
on a N^d Cartesian lattice in d-dimension, e.g., the exciton Bose liquid in two
dimension, we explicitly show that a belt subsystem with width L preserving
translational symmetry along d-1 Cartesian axes has leading entanglement
entropy (N^{d-1}/3)ln L. Using this result, the strong subadditivity
inequality, and lattice symmetries, we bound the entanglement entropy of a
rectangular subsystem from below and above showing a logarithmic violation of
the area law. For subsystems with a single flat boundary we also bound the
entanglement entropy from below showing a logarithmic violation, and argue that
the entanglement entropy of subsystems with arbitrary smooth boundaries are
similarly bounded.
