A class of two-dimensional AKLT models with a gap

Abstract

The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal lattice exhibits a spectral gap. In this paper, we introduce a family of variants of the two-dimensional AKLT model depending on a positive integer $n$, which is defined by decorating the edges of the hexagonal lattice with one-dimensional AKLT spin chains of length $n$. We prove that these decorated models are gapped for all $n\ge 3$.

arXiv commentvarious corrections, version accepted for publication in Contemporary Mathematics (AMS)

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Cited by in corpus (19)

• AKLT models on decorated square lattices are gapped
• Improved local spectral gap thresholds for lattices of finite dimension
• Spectral Gaps and Incompressibility in a $\nu =1/3$ Fractional Quantum Hall System
• Gaplessness is not generic for translation-invariant spin chains
• Efficient Verification of Ground States of Frustration-Free Hamiltonians
• Some aspects of Affleck-Kennedy-Lieb-Tasaki models: tensor network, physical properties, spectral gap, deformation, and quantum computation
• Quantitatively improved finite-size criteria for spectral gaps
• Random translation-invariant Hamiltonians and their spectral gaps
• Finite-size criteria for spectral gaps in $D$-dimensional quantum spin systems
• On the critical finite-size gap scaling for frustration-free Hamiltonians
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Entanglement subvolume law for 2D frustration-free spin systems
• Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model
• A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs
• The AKLT models on the singly decorated diamond lattice and two degree-4 planar lattices are gapped
• The Complexity of Approximating Critical Points of Quantum Phase Transitions
• On a bulk gap strategy for quantum lattice models
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• Order, Disorder, and Transitions in Decorated AKLT States on Bethe Lattices