We propose the ultra-fast numerical approach to large-scale inhomogeneous superconductors, which we call the Localized Krylov–Bogoliubov-de Gennes method (LK-BdG). In the LK-BdG method, the computational complexity of the local Green’s function, which is used to calculate the local density of states and the mean-fields, does not depend on the system size N. The calculation cost of self-consistent calculations is O(N). This method enables us to open a new avebue for treating extremely large systems with millions of lattice sites in the BdG framework. To show the power of the LK-BdG method, we demonstrate a self-consistent calculation of large-size Penrose quasicrystal lattice with a vortex. We also demonstrate that it takes less than 30 s with one CPU core on a laptop computer to calculate the local density of states with whole energy range in 100-millions-site tight-binding model.
 YN, J. Phys. Soc. Jpn. 89, 074703 (2020).
The caption: Elapsed time for calculating the LDOS at a vortex core. The calculations with a single CPU core are done on a laptop PC.
Keywords: Bogoliubov-de Gennes equations, inhomogeneous superconductor, vortex