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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.
The 1D AKLT model is a paradigm of antiferromagnetism, and its ground state exhibits symmetry-protected topological order. On a 2D lattice, the AKLT model has recently gained attention because it too displays symmetry-protected topological order, and
The $S=1$ Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first rigorous example of an isotropic spin system in the Haldane phase. The conjecture that the $S=3/2$ AKLT model on the hexagonal lattice is also in a gapped phase has remaine
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
The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable model
The quantum complexity of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational co