A Fractal Lattice Ising Model with Pair and Triplet Interactions

The equivalence of the Ising model on a fractal lattice having two- and three-site couplings with the Ising model having only two-site interaction in a non-zero magnetic field on the same lattice is derived by a partial tracing which effectively decimates triplet interaction. It is shown that the system is in ordered phase and no phase transition exists at any finite temperature. Only trivial transition occurs at T = ∞. In particular, we find that the pure triplet interaction Ising model is equivalent to one with pure nearest-neighbor pair interaction on the lattice, and the two systems have the same critical behavior. This is in contrast with that of triangular lattice and Sierpinski gasket fractal.