Corner and edge states in topological Sierpinski Carpet systems.

Fractal lattices, with their self-similar and intricate structures, offer potential platforms for engineering physical properties on the nanoscale and also for realizing and manipulating high order topological insulator states in novel ways. Here we present a theoretical study on localized corner and edge states, emerging from topological phases in Sierpinski Carpet within a $\pi$-flux regime. A topological phase diagram is presented correlating the quadrupole moment with different hopping parameters. Particular localized states are identified following spatial signatures in distinct fractal generations. The specific geometry and scaling properties of the fractal systems can guide the supported topological states types and their associated functionalities. A conductive device is proposed by coupling identical Sierpinski Carpet units providing transport response through projected edge states which carry on the details of the system's topology. Our findings suggest that fractal lattices may also work as alternative routes to tune energy channels in different devices.