Krylov localization as a probe for ergodicity breaking

Krylov complexity has recently gained attention where the growth of operator complexity in time is measured in terms of the off-diagonal operator Lanczos coefficients. The operator Lanczos algorithm reduces the problem of complexity growth to a single-particle semi-infinite tight-binding chain (known as the Krylov chain). Employing the phenomenon of Anderson localization, we propose the inverse localization length on the Krylov chain as a probe to detect weak ergodicity-breaking. On the Krylov chain we find delocalization in an ergodic regime, as we show for the SYK model, and localization in case of a weakly ergodicity-broken regime. Considering the dynamics beyond scrambling, we find a collapse across different system sizes at the point of weak ergodicity-breaking leading to a quantitative prediction. We further show universal traits of different operators in the ergodic regime beyond the scrambling dynamics. We test for two settings: (1) the coupled SYK model, and (2) the quantum East model. Our findings open avenues for mapping ergodicity/weak ergodicity-breaking transitions to delocalization/localization phenomenology on the Krylov chain.