Krylov Delocalization/Localization across Ergodicity Breaking

TL;DR

This paper explores how operator complexity growth relates to localization phenomena on the Krylov chain, revealing a transition between delocalization in ergodic regimes and localization in weakly ergodic regimes, with implications for understanding ergodicity breaking.

Contribution

It introduces a phenomenological framework linking ergodicity breaking to localization transitions on the Krylov chain, supported by analysis of the SYK and East models.

Findings

Delocalization occurs in the ergodic regime of the Krylov chain.

Localization occurs in weakly ergodic regimes.

Collapse of operator dynamics is observed in the ergodic regime.

Abstract

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 phenomenology of inverse localization length on the Krylov chain that undergoes delocalization/localization transition on the Krylov chain while the physical system undergoes 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 operators in the ergodic regime. 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.