Topological random fractals

Abstract The search for novel topological quantum states has recently moved beyond naturally occurring crystalline materials to complex and engineered systems. In this work we generalize the notion of electronic random lattices in non-integer dimensions. By considering a class D tight-binding model on critical clusters resulting from two-dimensional site percolation process, demonstrate that these fractals exhibit hallmarks insulators. Specifically, our large-scale numerical studies reveal display robust mobility gap, support quantized conductance represent well-defined thermodynamic phase matter. finite-size scaling analysis further suggests properties are not consistent with expectations systems two dimensions, hinting nontrivial relationship between fractal integer-dimensional states. Our results establish as most known band topology their distinct unique properties.