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Number theory, periodic orbits and superconductivity in nano-cubes

We study superconductivity in isolated superconducting nano-cubes and nano-squares of size $L$ in the limit of negligible disorder, $δ/Δ_0 \ll 1$ and $k_F L \gg 1$ for which mean-field theory and semiclassical techniques are applicable, with $k_F$ the Fermi wave vector, $δ$ the mean level spacing and $Δ_0$ the bulk gap. By using periodic orbit theory and number theory we find explicit analytical expressions for the size dependence of the superconducting order parameter. Our formalism takes into account contributions from both the spectral density and the interaction matrix elements in a basis of one-body eigenstates. The leading size dependence of the energy gap in three dimensions seems to be universal as it agrees with the result for chaotic grains. In the region of parameters corresponding to conventional metallic superconductors, and for sizes $L \gtrsim 10$nm, the contribution to the superconducting gap from the matrix elements is substantial ($\sim 20\%$). Deviations from the bulk limit are still clearly observed even for comparatively large grains $L \sim 50$nm. These analytical results are in excellent agreement with the numerical solution of the mean-field gap equation.