Contextuality in infinite one-dimensional translation-invariant local Hamiltonians

In recent years there has been a growing interest in treating many-body systems as Bell scenarios, where lattice sites play the role of distant parties and only near-neighbor statistics are accessible. We investigate contextuality arising from three Bell scenarios in infinite, translation-invariant 1D models: nearest-neighbor with two dichotomic observables per site; nearest- and next-to-nearest neighbor with two dichotomic observables per site, and nearest-neighbor with three dichotomic observables per site. For the first scenario, we give strong evidence that it cannot exhibit contextuality, not even in non-signaling physical theories beyond quantum mechanics. For the second one, we identify several low-dimensional models that reach the ultimate quantum limits, paving the way for self-testing ground states of quantum many-body systems. For the last scenario, which generalizes the Heisenberg model, we give strong evidence that, in order to exhibit contextuality, the dimension of the local quantum system must be at least 3.