Spectral gap of the AKLT model on the square lattice
In plain words
The AKLT model is a specially designed quantum magnet whose ground state is known exactly. Its energy gap is proven in one dimension and on the honeycomb lattice but not on the square lattice.
Precise statement
Spin-2 AKLT model on $Z^{2}$: H = sum_<ij> P^(4)_{ij}, with P^(4) the projector onto total spin 4 of a nearest-neighbour pair. Prove that the gap above the unique ground state is bounded below uniformly in system size. Answer: a proof, possibly via a finite-size criterion checked by computer.
What would settle it
A finite-size gap criterion verified numerically with rigorous error control, or an analytic proof.
Status in the literature
Gaps are proven on the hexagonal lattice (Lemm, Sandvik and Wang; Pomata and Wei, 2019-2020) and on decorated square lattices (2019), extended to generalized decorated graphs (Lucia and Young, 2023) and other spin-2 lattices (Guo, Pomata and Wei, 2021); the undecorated square lattice remains unproven.