An infinity of phase transitions as a function of temperature: exact results for a model with fixed-point imaging

Position-space renormalization-group methods are used to derive exact results for an Ising model on a fractal lattice. The model incorporates both nearest-neighbor and long-range interactions. The long-range interactions, which span all length scales on the lattice, can be thought of as resulting from fractal periodic boundary conditions. We present exact phase diagrams and specific heats in terms of these two interactions, and show that a “hall of mirrors” fixed-point imaging mechanism leads to an infinite number of phase transitions.