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We show the violation of the entanglement-area law for bosonic systems withBose surfaces. For bosonic systems with gapless factorized energy dispersionson a N^d Cartesian lattice in d-dimension, e.g., the exciton Bose liquid in twodimension, we explicitly show that a belt subsystem with width L preservingtranslational symmetry along d-1 Cartesian axes has leading entanglemententropy (N^{d-1}/3)ln L. Using this result, the strong subadditivityinequality, and lattice symmetries, we bound the entanglement entropy of arectangular subsystem from below and above showing a logarithmic violation ofthe area law. For subsystems with a single flat boundary we also bound theentanglement entropy from below showing a logarithmic violation, and argue thatthe entanglement entropy of subsystems with arbitrary smooth boundaries aresimilarly bounded.

Violation of Entanglement-Area Law in Bosonic Systems with Bose Surfaces: Possible Application to Bose Metals