Undecidability of the spectral gap in rotationally symmetric Hamiltonians

The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [] or more dimensions []. In these works, families of nearest-neighbor interactions are constructed whose spectral gap depends on the outcome of a Turing machine halting problem, therefore making it impossible for an algorithm to predict its existence. While these models are translationally invariant, they are not invariant under the other symmetries of the lattice, a property which is commonly found in physically relevant cases. This poses the question of whether the spectral gap problem could be decidable for Hamiltonians with stronger symmetry constraints. We give a negative answer to this question, in the case of models with four-body (plaquette) interactions on the square lattice satisfying rotation, but not reflection, symmetry: rotational symmetry is not enough to make the problem decidable.