Emergence of non-ergodic multifractal quantum states in geometric fractals

Eigenstate multifractality, a hallmark of non-interacting disordered metals, features anomalously slow dynamics and is relevant across quantum platforms ranging from measurement-driven systems to superconductivity. Despite extensive work on multifractality in disordered and quasiperiodic systems, deterministic, disorder-free lattice routes remain underexplored. Here, we demonstrate non-ergodic multifractal states (NEMs) by iteratively placing defects to shape a crystal lattice into a fractal geometry. Through a comprehensive analysis of the Sierpiński gasket, we find consistent evidence for the emergence of NEMs that go beyond the conventional classification of quantum states and open new pathways for quantum transport studies. We further outline experimental signatures of these states. Quantum fractals offer a clean route to unconventional states between a metal and an insulator. In a disorder-free Sierpiński gasket lattice, the authors characterise the emergence of non-ergodic multifractal states, predict experimental signatures, and suggest pathways toward precise quantum control.