  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.
