Exact solutions to plaquette Ising models with free and periodic boundaries

TL;DR

This paper provides exact solutions for the 2d and 3d plaquette Ising models under various boundary conditions, clarifying their partition functions and revealing how boundary choices influence correlations in finite systems.

Contribution

It establishes the precise relationship between free and periodic boundary conditions in plaquette Ising models and solves the 2d model with helical boundaries, enhancing understanding of boundary effects.

Findings

Partition functions relate differently under free and periodic boundaries.

Boundary conditions induce correlations in finite plaquette Ising systems.

Exact solutions for 2d plaquette models with helical boundaries are provided.

Abstract

An anisotropic limit of the 3d plaquette Ising model, in which the plaquette couplings in one direction were set to zero, was solved for free boundary conditions by Suzuki (Phys. Rev. Lett. 28 (1972) 507), who later dubbed it the fuki-nuke, or "no-ceiling", model. Defining new spin variables as the product of 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 boundary conditions, which require a different approach to defining the product spin transformation, by Castelnovo et al. (Phys. Rev. B 81 (2010) 184303). We clarify the exact relation between partition functions with free and periodic boundary conditions expressed in terms of original and product spin variables for the 2d plaquette and 3d fuki-nuke models, noting that the differences are already present in the 1d Ising model. In addition, we solve the 2d plaquette Ising model with helical boundary conditions. The various exactly solved examples illustrate how correlations can be induced in finite systems as a consequence of the choice of boundary conditions.