A numerical study of entanglement measures of clusters joined by a point contact

The ground state of an antiferromagnetic Heisenberg model on L X L clusters joined by a single bond and balanced Bethe clusters are investigated with quantum Monte Carlo. The improved Monte Carlo method of Sandvik and Evertz is used and the observables include valence bond and loop valence bond observables introduced by Lin and Sandvik as well as the valence bond entropy and the second Renyi entropy. For the bisecting of the Bethe cluster, in disagreement with our previous results and in agreement with mean field theory, the valence loop entropy and the second Renyi entropy scale as the log of the number of sites in the cluster. For bisecting the L X L - LX L clusters, all the observables appear to scale as L in agreement with our previous results. This is in disagreement with a straight forward application of the area law.