Boundary effects to the entanglement entropy and two-site entanglement   of the spin-1 valence-bond solid
  We investigate the von Neumann entropy of a block of subsystem for the valence-bond solid (VBS) state with general open boundary conditions. We show that the effect of the boundary on the von Neumann entropy decays exponentially fast in the distance between the subsystem considered and the boundary sites.   Further, we show that as the size of the subsystem increases, its von Neumann entropy exponentially approaches the summation of the von Neumann entropies of the two ends, the exponent being related to the size. In contrast to critical systems, where boundary effects to the von Neumann entropy decay slowly, the boundary effects in a VBS, a non-critical system, decay very quickly. We also study the entanglement between two spins.   Curiously, while the boundary operators decrease the von Neumann entropy of L spins, they increase the entanglement between two spins. 
Heng Fan, Vladimir Korepin, Vwani Roychowdhury, Christopher Hadley,   and Sougato Bose