We present the (numerically) exact phase diagram of a magnetic polymer on the Sierpińsky gasket embedded in three dimensions using the renormalization group method. We report distinct phases of the magnetic polymer, including paramagnetic swollen, ferromagnetic swollen, paramagnetic collapsed, and ferromagnetic collapsed states. By evaluating critical exponents associated with phase transitions, we located the phase boundaries between different phases. If the model is extended to include a four-site interaction which disfavors configurations with a single spin of a given type, we find a rich variety of critical behaviors. Notably, we uncovered a phenomenon of reentrance, where the system transitions from a collapsed (paramagnetic) state to a swollen (paramagnetic) state followed by another collapse (paramagnetic) and ultimately reaching a ferromagnetic collapsed state. These findings shed new light on the complex behavior of (lattice) magnetic polymers.

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• Phase diagrams
• Renormalization group
• Self-avoiding walks
• Critical phenomena
• Ising model