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The AKLT model on a hexagonal chain is gapped
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In 1987, Affleck, Kennedy, Lieb, and Tasaki introduced the AKLT spin chain and proved that it has a spectral gap above the ground state. Their concurrent conjecture that the two-dimensional AKLT model on the hexagonal lattice is also gapped remains open. In this paper, we show that the AKLT Hamiltonian restricted to an arbitrarily long chain of hexagons is gapped. The argument is based on explicitly verifying a finite-size criterion which is tailor-made for the system at hand. We also discuss generalizations of the method to the full hexagonal lattice.
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Improved local spectral gap thresholds for lattices of finite dimension
For frustration-free Hamiltonians on any finite-dimensional lattice, the minimum spectral gap of any rectangular region is O(γ + 1/t²) where t is the shortest side length, improving previous thresholds.
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