Entanglement in Valence-Bond-Solid States

Abstract

This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Renyi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.

Published version, 80 pages, 8 figures; references updated

References in corpus (22)

• Entanglement detection
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Diverging Entanglement Length in Gapped Quantum Spin Systems
• Valence Bond Solids for Quantum Computation
• Entanglement entropy in fermionic Laughlin states
• Measurement-based quantum computation beyond the one-way model
• Implementation of Spin Hamiltonians in Optical Lattices
• Measurement-based quantum computer in the gapped ground state of a two-body Hamiltonian
• Valence bond solids for SU(n) spin chains: exact models, spinon confinement, and the Haldane gap
• Exact results for SU(3) spin chains: trimer states, valence bond solids, and their parent Hamiltonians
• Entanglement of Two Impurities through Electron Scattering
• Exact Analysis of Entanglement in Gapped Quantum Spin Chains
• Spinon confinement and the Haldane gap in SU(n) spin chains
• Entanglement in an SU(n) Valence-Bond-Solid State
• Entropy and Exact Matrix Product Representation of the Laughlin Wave Function
• Geometric Entanglement in a One-Dimensional Valence Bond Solid State
• Entanglement and Density Matrix of a Block of Spins in AKLT Model
• Spontaneous Parity Violation in a Quantum Spin Chain
• Boundary effects to the entanglement entropy and two-site entanglement of the spin-1 valence-bond solid
• 2D Multipartite Valence Bond States in Quantum Antiferromagnets
• Entanglement of Valence-Bond-Solid on an Arbitrary Graph
• Block Spin Density Matrix of the Inhomogeneous AKLT Model

Cited by in corpus (22)

• Entanglement of Exact Excited States of AKLT Models: Exact Results, Many-Body Scars and the Violation of Strong ETH
• Exact Excited States of Non-Integrable Models
• Phases of one-dimensional SU(N) cold atomic Fermi gases --from molecular Luttinger liquids to topological phases
• Google in a Quantum Network
• Candidate Quantum Spin Liquid in the Ce\textsuperscript{3+} Pyrochlore Stannate Ce${}_2$Sn${}_2$O${}_7$
• Measurement-induced steering of quantum systems
• Symmetry-protected topological phases of alkaline-earth cold fermionic atoms in one dimension
• Entanglement Spectra of the 2D AKLT Model: VBS/CFT Correspondence
• Entanglement as geometry and flow
• Discrete-time quantum walk algorithm for ranking nodes on a network
• Entanglement and correlation functions of the quantum Motzkin spin-chain
• Kitaev honeycomb tensor networks: exact unitary circuits and applications
• Entanglement and SU(n) symmetry in one-dimensional valence bond solid states
• Measurement-Based Quantum Computing with Valence-Bond-Solids
• Efficient verification of Affleck-Kennedy-Lieb-Tasaki states
• Efficient Verification of Ground States of Frustration-Free Hamiltonians
• Bulk-Edge correspondence of entanglement spectrum in 2D spin ground states
• Spin-spin correlation functions of the q-VBS state of an integer spin model
• Negativity in the Generalized Valence Bond Solid State
• Entanglement of a alternating bipartition in spin chains: Relation with classical integrable models
• Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
• Angular-time evolution for the Affleck-Kennedy-Lieb-Tasaki chain and its edge-state dynamics