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 ≥ 3.

AKLT models on decorated square lattices are gapped

“…We note that the results of Ref. [1], as argued below, apply directly to other trivalent lattices with decoration, such as the square-octagon (4, 8 2 ), the cross (4,6,12), and the star (3, 12 2 ); see e.g. Fig.…”

“…Here we briefly review the key points that enable the proof of the spectral gap for AKLT models on the dec-orated honeycomb lattice in Ref. [1]; see Fig. 1a for one such illustration with n = 1, as well as other lattices.…”

“…Regarding the gap, tensor network methods were employed and the value of the gap in the thermodynamic limit was estimated [18,19]. A recent breakthrough in the analytic proof was given by Abdul-Rahman et al [1], who considered a family of decorated honeycomb lattices and proved that the corresonding AKLT models are gapped for the number n of decorated sites being greater than 2; see e.g. Fig.…”

Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice

“…AKLT also investigated the S = 3/2 model on the hexagonal lattice and were able to demonstrate the exponential decay of the spin-spin correlations for the exact VBS ground state with periodic boundary conditions, and on the basis of this fact they conjectured that the hexagonal model also exhibits a spectral gap (see also [22]). Evidence pointing to a spectral gap has been mounting [22][23][24][25][26][27][28], but, despite the paradigmatic role played by the hexagonal AKLT model, the foundational question of whether its spectrum is gapped has remained open. The presence of a gap would have wider consequences, e.g., in supporting the heuristic that PEPS arise from gapped Hamiltonians [29].…”

“…More generally, the existing mathematical techniques for deriving spectral gaps in quantum spin systems of dimensions ≥ 2 are quite limited. The few examples where a spectral gap is known to exist include the product vacua with boundary states (PVBS) models [30][31][32] and, since recently, decorated variants of the AKLT models [23,28].…”