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An anisotropic limit of the 3d plaquette Ising model, in which the plaquettecouplings in one direction were set to zero, was solved for free boundaryconditions by Suzuki (Phys. Rev. Lett. 28 (1972) 507), who later dubbed it thefuki-nuke, or "no-ceiling", model. Defining new spin variables as the productof nearest-neighbour spins transforms the Hamiltonian into that of a stack of(standard) 2d Ising models and reveals the planar nature of the magnetic order,which is also present in the fully isotropic 3d plaquette model. More recently,the solution of the fuki-nuke model was discussed for periodic boundaryconditions, which require a different approach to defining the product spintransformation, by Castelnovo et al. (Phys. Rev. B 81 (2010) 184303).  We clarify the exact relation between partition functions with free andperiodic boundary conditions expressed in terms of original and product spinvariables for the 2d plaquette and 3d fuki-nuke models, noting that thedifferences are already present in the 1d Ising model. In addition, we solvethe 2d plaquette Ising model with helical boundary conditions. The variousexactly solved examples illustrate how correlations can be induced in finitesystems as a consequence of the choice of boundary conditions.

Exact solutions to plaquette Ising models with free and periodic boundaries