N-independent Localized Krylov–Bogoliubov-de Gennes Method: Ultra-fast Numerical Approach to Large-scale Inhomogeneous Superconductors

Abstract

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 \(\mathcal{O}(N)\), which enables us to open a new avenue for treating extremely large systems with millions of lattice sites. To show the power of the LK-BdG method, we demonstrate a self-consistent calculation on the 143806-site Penrose quasicrystal lattice with a vortex and a calculation on 1016064-site two-dimensional nearest-neighbor square-lattice tight-binding model with many vortices. We also demonstrate that it takes less than 30 s with one CPU core to calculate the local density of states with whole energy range in 100-millions-site tight-binding model.