  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 modified spin-wave 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
modified spin-wave theory, the valence loop entropy and the second Renyi
entropy scale as the logarithm of the number of sites in the cluster. For
bisecting the L X L - L X L clusters, the valence bond entropy scales as L,
however, the loop entropy and the entanglement entropy scale as ln(L). For the
entanglement entropy, the coefficient of the logarithm, the number of Goldstone
modes/2, is universal and is the analogue of the sub leading term of a LXL
cluster. This result was substantiated by a modified spin-wave theory
calculation for the ferromagnetic X-Y model. Taken together, the calculations
suggest that linking high entanglement objects will not generate much more
entanglement. As a consequence, simulating the ground state of clusters joined
together by a few bonds should not be much more difficult than simulating the
ground state of a single cluster.
