Mode engineering in reconfigurable fractal topological circuits

Circuits can provide a versatile platform for exploring new physics, particularly in probing the topological phases within complex geometries. Fractals, celebrated for their intricate, self-similar duality, and noninteger dimensions, particularly those embedded in complex manifolds, remain uncharted in this context. In our research, we implement Sierpi {n} \'{n}{}ski fractal topological insulators within reconfigurable fractal topological circuits while expanding the scope to include the cylindrical and toroidal structures. Our approach is grounded in consistency theory and reinforced through experimental verification, confirming the presence of unconventional higher-order topological phenomena referring to the abundance of topological edge and corner modes. Intriguingly, the quantity of these edge and corner modes is proportional to the volume modes relative to the system size, with an exponent aligning with the Hausdorff fractal dimension of the Sierpi {n} \'{n}{}ski carpet. This study paves the way for a deeper exploration of topological modes within fractal geometries, potentially unlocking new avenues in topological physics.