Entanglement in valence-bond-solid states on symmetric 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.

32 pages, 17 figures, 4 tables

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• Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource
• Quantum computational capability of a 2D valence bond solid phase
• Entanglement Spectra of the 2D AKLT Model: VBS/CFT Correspondence
• Universal measurement-based quantum computation with spin-2 Affleck-Kennedy-Lieb-Tasaki states
• Entanglement Spectra of the quantum hard-square model: Holographic minimal models
• Topological and dynamical properties of a generalized cluster model in one dimension
• Quantum many-body scars in spin models with multibody interactions
• Entanglement spectra of the q-deformed Affleck-Kennedy-Lieb-Tasaki model and matrix product states
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• The matrix product representation for the q-VBS state of one-dimensional higher integer spin model
• How localized bosons manage to become superfluid
• Bulk-Edge correspondence of entanglement spectrum in 2D spin ground states
• Entanglement signatures for the dimerization transition in the Majumdar-Ghosh model
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• A useful way to obtain the central charge of entanglement Hamiltonian -- Nested entanglement entropy
• A brief note on the G${}_2$ Affleck-Kennedy-Lieb-Tasaki chain
• Vestigial Nematic Order at Zero Temperature in Two-Dimensional Frustrated Quantum Antiferromagnets