This paper shows that certain quantum spin models introduced by Affleck, Kennedy, Lieb and Tasaki (AKLT) satisfy a property called Local Topological Quantum Order (LTQO) on the hexagonal lattice and on a decorated square lattice known as the Lieb lattice. In plain terms, the authors prove that ground states computed on large finite patches of these lattices become indistinguishable from a single infinite-system ground state whenever the observable is far from the patch boundary. The error in this approximation decays exponentially with that distance.

AKLT models are frustration-free nearest-neighbor quantum spin systems. In the cases studied here, the hexagonal model has spin-3/2 degrees of freedom at the original lattice sites. The Lieb lattice is a square lattice with an extra site added on each edge; in that model the degree-4 sites carry spin-2 and the degree-2 decorating sites carry spin-1. The paper also treats families of “decorated” lattices obtained by inserting m extra sites on each edge; the LTQO result holds for all decorations on the hexagonal lattice and for decorations with m ≥ 1 on the square-derived Lieb family.

To get these results the authors identify a convenient sequence of growing finite volumes and then compare finite-volume ground-state expectations with a specific infinite-volume ground state. They show the difference is bounded by a term that falls off exponentially with the distance from the observable to the boundary of the finite volume. The proof builds on an older “polymer representation” of the AKLT ground state due to Kennedy, Lieb and Tasaki (1988). The authors adapt that representation and then use cluster-expansion tools and convergence criteria of Kotecký–Preiss and Ueltschi to control the sums that appear.

LTQO is important because it is a central hypothesis in general mathematical strategies for proving stability of a spectral gap. The spectral gap is the energy difference between the ground state and the first excited state; a nonzero gap is a hallmark of a stable phase and controls properties such as decay of correlations. As a corollary of LTQO, the paper shows that the spectral gap above the ground state in these AKLT models is stable under general small perturbations that decay sufficiently fast in space. The authors also use the same cluster-expansion methods to deduce exponential decay of two-point correlations in the class of models they consider.