Observing Magnetic Order on Fractal Lattices via Quantum Annealing

We realize field-tunable Ising magnetism on finite-generation Sierpiński lattices using a D-Wave quantum annealer, enabling controlled tests of cooperative order on a fractal graph. Field sweeps drive shell-selective reorganizations that depart from the behavior of periodic triangular lattices. Momentum-space structure factors show sixfold Bragg-like features fixed by the fractal scaffold and a hierarchy of self-similar satellites. Decomposing the structure factor cleanly separates immutable form factor from correlations. Peak positions are geometry-locked, while satellite intensities are reweighted generation-by-generation and collapse toward the pure form-factor limit as polarization sets in. These results provide a materials-relevant framework for identifying hierarchical order and frustration in artificial spin-ice arrays and other patterned magnetic metamaterials.