Boundary conditions dependence of the phase transition in the quantum Newman-Moore model

We study the triangular plaquette model (TPM, also known as the Newman-Moore model) in the presence of a transverse magnetic field on a lattice with periodic boundaries in both spatial dimensions. We consider specifically the approach to the ground state phase transition of this quantum TPM (QTPM, or quantum Newman-Moore model) as a function of the system size and type of boundary conditions. Using cellular automata methods, we obtain a full characterization of the minimum energy configurations of the TPM for arbitrary tori sizes. For the QTPM, we use these cycle patterns to obtain the symmetries of the model, which we argue determine its quantum phase transition: we find it to be a first-order phase transition, with the addition of spontaneous symmetry breaking for system sizes which have degenerate classical ground states. For sizes accessible to numerics, we also find that this classification is consistent with exact diagonalization, Matrix Product States and Quantum Monte Carlo simulations.