Entanglement in valence-bond-solid states on symmetric graphs

TL;DR

This paper investigates quantum entanglement in the valence-bond-solid ground state of the AKLT model on symmetric 2D graphs, revealing deviations from naive entropy expectations due to hidden boundary spin chains.

Contribution

It introduces a numerical method to analyze entanglement in AKLT states on symmetric graphs, highlighting boundary effects and hidden spin chains.

Findings

Eigenvalues of reduced density matrices obtained via Monte Carlo integration.

Deviation from expected von Neumann entropy per valence bond.

Analytical confirmation of results on ladder graphs.

Abstract

We study quantum entanglement in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model defined on two-dimensional graphs with reflection and/or inversion symmetry. The ground state of this spin model is known as the valence-bond-solid state. We investigate the properties of reduced density matrix of a subsystem which is a mirror image of the other one. Thanks to the reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is $\ln 2$. This deviation is interpreted in terms of the hidden spin chain along the boundary between the subsystems. In some cases where graphs are on ladders, the numerical results are analytically or algebraically confirmed.