Topological fractal braiding of non-Hermitian bands

Braiding is one of the oldest crafting techniques in human history, yet today it plays a key role across a wide range of scientific disciplines, from biology to physics. However, how fractal geometry interplays with non-Hermitian band braiding remains unexplored. Here, we bridge fractal physics and non-Hermitian topology by introducing topological fractal braiding, which can produce self-similar topological winding matrices. Furthermore, through tight-binding lattice model construction, we prove these fractal winding matrices govern non-Hermitian skin modes, yielding a scale-invariant skin effect where the ratio of left- to right-localized skin modes remains conserved through fractal iterations. We design and fabricate reconfigurable non-Hermitian topoelectrical circuits to experimentally reconstruct the fractal braiding of non-Hermitian bands, confirming our theory. Our work resolves how fractal geometry manifests in non-Hermitian band braiding and impacts skin effects, demonstrating scalable control of complex spectral topologies. While non-Hermitian physics and fractal geometry have been widely studied, their interplay in band braiding remains largely unexplored. The authors establish topological fractal braiding in non-Hermitian systems with scale-invariant skin effects, experimentally demonstrated using reconfigurable circuits.