Krylov fractality and complexity in generic random matrix ensembles

Krylov space methods provide an efficient framework for analyzing the static and dynamical aspects of quantum systems, with tridiagonal matrices playing a key role. Despite their importance, the behavior of such matrices from chaotic to integrable states, transitioning through an intermediate phase, remains unexplored. We aim to fill this gap by considering the properties of the tridiagonal matrix elements and the associated basis vectors for appropriate random matrix ensembles. We utilize the Rosenzweig-Porter model as our primary example, which hosts a fractal phase in addition to the ergodic and localized regimes. We discuss the characteristics of the matrix elements and basis vectors across these three (ergodic, fractal, and localized) regimes and introduce Krylov space tools to identify the transition points. The exact expressions of the Lanczos coefficients of such tridiagonal matrices are provided in terms of $q$-logarithmic function across the full parameter regime. The numerical results are corroborated with analytical reasoning for certain features of the Krylov spectra. Additionally, we investigate the Krylov state (spread) complexity within these regimes, showcasing the efficacy of our methods in pinpointing these transitions.