Krylov formulation of the Numerical Renormalization Group

We present a Krylov strategy for greatly reducing the computational costs of the Numerical Renormalization Group (NRG) calculations for quantum impurity models. Standard Wilsonian NRG uses expensive brute force exact diagonalization to compute the full spectrum of each Wilson shell; then, high-energy states discarded, while low-energy states are kept to compute the next shell. By constrast, Krylov NRG (KNRG) constructs only kept states, obtained by building a Krylov space just large enough to ensure an accurate description of the next shell. Discarded states are not computed at all during the forward sweep. Instead, those needed for an accurate description of dynamics, e.g. when computing spectral functions, are computed on the fly during a backward sweep, using a Krylov scheme targeting the discarded sector. KNRG yields accurate results and fulfills spectral sum rules by construction. By focusing on only the most relevant states of each Wilson shell instead of resolving it fully, KNRG reduces both computation times and memory very significantly.