Random translation-invariant Hamiltonians and their spectral gaps

Quantum 6, 790 (2022).

https://doi.org/10.22331/q-2022-09-01-790

We consider random translation-invariant frustration-free quantum spin Hamiltonians on $mathbb Z^D$ in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.