Topological estimator of block entanglement for Heisenberg antiferromagnets

We introduce a computable estimator of block entanglement entropy for many-body spin systems admitting total singlet ground states. Building on a simple geometrical interpretation of entanglement entropy of the so-called valence bond states, this estimator is defined as the average number of common singlets to two subsystems of spins. We show that our estimator possesses the characteristic scaling properties of the block entanglement entropy in critical and noncritical one-dimensional Heisenberg systems. We invoke this new measure to examine entanglement scaling in the two-dimensional Heisenberg model on a square lattice revealing an "area law" for the gapped phase and a logarithmic correction to this law in the gapless phase.