Gonihedric Ising Actions

where the sum is over the edges of some triangulated surface, θ(αij) = |π − αij | ζ , ζ is some exponent, and αij is the dihedral angle between neighbouring triangles with common link < ij >. This definition of the action was inspired by the geometrical notion the linear size of a surface, originally defined by Steiner. In equ.(1) it is the surface itself that is discretized, rather than the space in which it is embedded. An alternative approach to discretizing the linear size is to discretize the embedding space by restricting the allowed surfaces to the plaquettes of a (hyper)cubic lattice. This method was applied by Savvidy and Wegner [2– 5], who rewrote the resulting theory as a generalized Ising model by using the geometrical spin cluster boundaries to define the surfaces. The energy of a surface on a cubic lattice is given explicitly in the Savvidy-Wegner models by E = n2 +4κn4, where n2 is the number of links where