  A 3D Ising model with a purely plaquette, 4-spin interaction displays a
planar flip symmetry intermediate between a global and a gauge symmetry and as
a consequence has a highly degenerate low temperature phase and no standard
magnetic order parameter. This plaquette Hamiltonian is a particular case of a
family of 3D Gonihedric Ising models defined by Savvidy and Wegner. An
anisotropic variant of the the purely plaquette Gonihedric model, originally
discussed as the "Fuki-Nuke" model by Suzuki, is non-trivially equivalent to a
stack of 2D Ising models, each of which can magnetize independently at the
phase transition point.
  Consideration of this anisotropic model suggests that a suitable order
parameter in the isotropic case may also be constructed using a form of planar
magnetization, in which nearest neighbour correlators summed over planes
replace a sum over spin values. We conduct Monte-Carlo simulations to
investigate this and related candidate order parameters in a toy model of
Gonihedric ground states, the Fuki-Nuke model and the isotropic plaquette
Gonihedric model itself.
