Demonstrating the Affleck-Kennedy-Lieb-Tasaki Spectral Gap on 2D Degree-3 Lattices

Abstract

We establish that the spin-3/2 AKLT model on the honeycomb lattice has a nonzero spectral gap. We use the relation between the anticommutator of two projectors and their sum, and apply it to related AKLT projectors that occupy plaquettes or other extended regions. We analytically reduce the complexity in the resulting eigenvalue problem and use a Lanczos numerical method to show that the required inequality for the nonzero spectral gap holds. This approach is also successfuly applied to several other spin-3/2 AKLT models on degree-3 semiregular tilings, such as the squareoctagon, star and cross lattices, where the complexity is low enough that exact diagonalization can be used instead of the Lanczos method. In addition, we also close the previously open cases in the singly decorated honeycomb and square lattices.

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.…”