Entanglement in the ground state of clusters joined by a single bond

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.