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.

Figure 1

A collection of singlets (a valence bond state) covering an exemplary lattice. The entanglement of the distinguished subsystem with its complement is the number of singlets crossing the boundary.

Figure 2

Types of lattices considered below: (a) the $J-J^{\prime }$ one-dimensional chain, (b) the ladder, (c) the $J-J^{\prime }$ staggered square lattice, (d) the $J-J^{\prime \prime }$ columnar square lattice. All interactions are assumed to be antiferromagnetic $J$, $J^{\prime }>0$. Periodic boundary conditions are assumed throughout.

Figure 3

(a) Scaling of ${\overline{\mathcal{N}}}_L$ for the critical chain with respect to the effective length $x(L)$. Inset: Comparison of the estimator and exact entanglement for the same partitions in a small system. (b) Saturation of ${\overline{\mathcal{N}}}_L$ for noncritical systems: diamonds—dimerized chain ($J^{\prime }/J=0.5$); triangles—spin ladder.

Figure 4

Scaling of the entanglement estimator per boundary area on the two-dimensional square lattice. Shown are data points for $J^{\prime }/J=1.0$, 6.0, 2.46, and $J^{\prime \prime }/J=1.91$. Insets: The estimator versus exact entanglement for small systems.