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

Abstract

The S = 1 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 remained open, despite being a fundamental question of ongoing relevance to condensed-matter physics and quantum information theory. Here we confirm this conjecture by demonstrating the size-independent lower bound ∆ > 0.006 on the spectral gap of the hexagonal model with periodic boundary conditions in the thermodynamic limit. Our approach consists of two steps combining mathematical physics and high-precision computational physics. We first prove a mathematical finite-size criterion which gives a rigorous, size-independent bound on the spectral gap if the gap of a particular cut-out subsystem of 36 spins exceeds a certain threshold value. Then we verify the finite-size criterion by performing state-of-the-art DMRG calculations on the subsystem.

Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

“…That result, however, has only limited bearing on what we can learn mathematically for specific classes of systems. There are a number of examples in the literature of such systems for which the question has been settled [1,11,16,43,44,[64][65][66]91,92]. For one-dimensional frustration-free systems arguments to prove a gap have been extended even further [2,23,39,59,61,62,70,77,84,98].…”

Measurement-Based Quantum Computation

“…AKLT showed that the spatial correlation function in the ground state decays exponentially, but the existence of the gap could not be proved (Affleck et al, 1987). Recently, two groups independently used numerically assisted approaches to show that the AKLT model indeed possesses a nonzero spectral gap (Lemm et al, 2020;Pomata & Wei, 2020), even in the limit that the system size becomes infinite. Therefore, the AKLT models provide example Hamiltonians that are shortranged, gapped, and have a unique ground state that is universal for measurement-based quantum computation.…”

Quantitatively improved finite-size criteria for spectral gaps

“…Motivated by the AKLT conjecture that the honeycomb AKLT model exhibits a spectral gap [2], Knabe investigated a finite-size criterion on the honeycomb lattice already in 1988 [12], albeit inconclusively. Recent years have seen a number of applications of finite-size criteria to the honeycomb and decorated honeycomb lattices [1,18,22,23], in some cases combine with extensive numerical calculations. Here we derive for the first time a general finite-size criterion of the form (2.1) for the honeycomb lattice which has the expected inverse-square threshold scaling just like the Euclidean case studied in Theorem 2.2 but naturally with different constants.…”

“…Starting with the work of Gosset-Mozgunov [9], finite-size criteria have seen considerable progress in the last years, especially in the context of higher-dimensional quantum spin systems. This progress concerns both the methodological side [3,15,16] and the applications side [1,13,14,17,18,21,22,23,25]. Recent methodological advances have mostly focused on the scaling of the gap threshold (called t ℓ in Eq.…”