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

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.