We show the violation of the entanglement area law for bosonic systems with Bose surfaces. For bosonic systems with gapless factorized energy dispersions on an $N^d$ Cartesian lattice in $d$ dimensions, e.g., the exciton Bose liquid in two dimensions, 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.

• Received 11 June 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.210402