Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

TL;DR

This paper compares valence bond and von Neumann entanglement entropies in Heisenberg systems, revealing their differences in 1D and 2D geometries and confirming the area law in two dimensions.

Contribution

It provides a direct comparison of two entanglement measures in Heisenberg ladders, highlighting their distinct behaviors and confirming the area law in 2D systems.

Findings

Valence bond entropy can be less or greater than von Neumann entropy in 1D.

In ladder geometries, von Neumann entropy obeys an area law.

Valence bond entropy shows a multiplicative logarithmic correction.

Abstract

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.