Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-$1/2$ Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.

Figure 1

Entanglement entropies for a 1D Heisenberg chain with PBC and OBC. Upper panels show the entropies as a function of the conformal distance $x^{\prime }=(L/\pi )\sin (\pi x/L)$ for 100-site chains. Lower plots show the central charge $c$, obtained by fitting the numerical data to the CFT result, for several $L$. For PBC, $c$ is calculated with the two smallest $x^{\prime }$ points removed. For OBC, the fits depend on the number of sites included $z$, which we systematically decrease by removing $x^{\prime }$ data points from the outside ends of the chain. $c$ is shown for $S^{\mathrm{vN}}$ (closed symbols) and $S^{\mathrm{VB}}$ (open symbols) for system sizes $L=64$ (circles), $L=100$ (squares), $L=128$ (diamonds), and $L=200$ (triangles)

Figure 2

Entanglement entropies for three-leg (left) and four-leg (right) ladder systems with OBC and 100 sites per leg. For odd-leg ladders, $S(x)\propto \ln (x^{\prime })$. The left-hand inset shows $S(x)$ as a function of the conformal distance, $x^{\prime }$, on a log scale. For even-leg ladders, $S(x\gtrsim \xi )=\mathrm{const}$. The right-hand inset shows the site indexing used for multileg ladders where the bipartition $A$ is shaded and labeled by $x=9$.

Figure 3

Entanglement entropies divided by $N$, for $N$-leg Heisenberg (filled symbols) and free-fermion (open diamonds) ladders, taken such that the region $A$ includes $2N^2$ sites. For the Heisenberg model and large $N$, $S^{\mathrm{VB}}\propto N\ln N$, whereas $S^{\mathrm{vN}}\propto N$. Data for free fermions, $S_{\mathrm{ff}}^{\mathrm{vN}}\propto N\ln N$, are shown for comparison. We show data for ladders with length (sites per leg) $L=100$ and $L=4N$.