  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 and mean field theory. 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 - L X 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.
