AKLT models on decorated square lattices are gapped

Abstract

The nonzero spectral gap of the original two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) models has remained unproven for more than three decades. Recently, Abdul-Rahman et al. [arXiv:1901.09297] provided an elegant approach and proved analytically the existence of a nonzero spectral gap for the AKLT models on the decorated honeycomb lattice (for the number $n$ of spin-1 decorated sites on each original edge no less than 3). We perform calculations for the decorated square lattice and show that the corresponding AKLT models are gapped if $n\ge 4$. Combining both results, we also show that a family of decorated hybrid AKLT models, whose underlying lattice is of mixed vertex degrees 3 and 4, are also gapped for $n\ge 4$. We develop a numerical approach that extends beyond what was accessible previously. Our numerical results further improve the nonzero gap to $n\ge 2$, including the establishment of the gap for $n=2$ in the decorated triangular and cubic lattices. The latter case is interesting, as this shows the AKLT states on the decorated cubic lattices are not Néel ordered, in contrast to the state on the undecorated cubic lattice.

arXiv comment11 pages, 7 figures, 2 tables

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Cited by in corpus (13)

• Demonstrating the AKLT spectral gap on 2D degree-3 lattices
• Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice
• Efficient verification of Affleck-Kennedy-Lieb-Tasaki states
• 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
• Random translation-invariant Hamiltonians and their spectral gaps
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
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• Deformations of the Boundary Theory of the Square Lattice AKLT Model
• Order, Disorder, and Transitions in Decorated AKLT States on Bethe Lattices