Demonstrating the AKLT spectral gap on 2D degree-3 lattices

Abstract

We establish that the spin-3/2 AKLT model on the honeycomb has a nonzero spectral gap. We use the relation between the anticommutator of two projectors and their sum, and apply it to related AKLT projectors that occupy plaquettes or other extended regions. We analytically reduce the complexity in the resulting eigenvalue problem and use a Lanczos numerical method to show that the required inequality for the nonzero spectral gap holds. This approach is also successfuly applied to several other spin-3/2 AKLT models on degree-3 semiregular tilings, such as the square octagon, star and cross lattices, where the complexity is low enough that exact diagonalization can be used instead of the Lanczos method. In addition, we also close the previously open cases in the singly decorated honeycomb and square lattices.

arXiv comment8 pages, 6 figures, 1 table. Minor revisions

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Cited by in corpus (22)

• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice
• Measurement-Based Quantum Computation
• Theory of triangulene two-dimensional crystals
• Orbital susceptibility of T-graphene: Interplay of high-order van Hove singularities and Dirac cones
• Preparing Valence-Bond-Solid states on noisy intermediate-scale quantum computers
• Spectral Gaps and Incompressibility in a $\nu =1/3$ Fractional Quantum Hall System
• Efficient verification of Affleck-Kennedy-Lieb-Tasaki states
• Undecidability of the Spectral Gap (full version)
• Some aspects of Affleck-Kennedy-Lieb-Tasaki models: tensor network, physical properties, spectral gap, deformation, and quantum computation
• Efficient Verification of Ground States of Frustration-Free Hamiltonians
• Quantitatively improved finite-size criteria for spectral gaps
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• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Quantum phases of a frustrated spin-1 system: The 5/7 skewed ladder
• Phase diagram of the SU($N$) antiferromagnet of spin $S$ on a square lattice
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• A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs
• Haldane phases and phase diagrams of the S = 3/2, 1 bilinear-biquadratic Heisenberg model on the orthogonal dimer chain
• The AKLT models on the singly decorated diamond lattice and two degree-4 planar lattices are gapped
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• Order, Disorder, and Transitions in Decorated AKLT States on Bethe Lattices