Strong enhancement of superconductivity on finitely ramified fractal lattices

Using the Sierpinski gasket (triangle) and carpet (square) lattices asexamples, we theoretically study the properties of fractal superconductors. Forthat, we focus on the phenomenon of $s$-wave superconductivity in the Hubbardmodel with attractive on-site potential and employ the Bogoliubov-de Gennesapproach and the theory of superfluid stiffness. For the case of the Sierpinskigasket, we demonstrate that fractal geometry of the underlying crystallinelattice can be strongly beneficial for superconductivity, not only leading to aconsiderable increase of the critical temperature $T_c$ as compared to theregular triangular lattice but also supporting macroscopic phase coherence ofthe Cooper pairs. In contrast, the Sierpinski carpet geometry does not lead topronounced effects, and we find no substantial difference as compared with theregular square lattice. We conjecture that the qualitative difference betweenthese cases is caused by different ramification properties of the fractals.