Krylov-space anatomy and spread complexity of a disordered quantum spin chain

TL;DR

This paper analyzes the structure and complexity of quantum states in Krylov space for disordered spin chains, revealing distinct behaviors in ergodic and many-body localized phases through complexity growth and eigenstate contributions.

Contribution

It introduces Krylov space spread complexity as a phase-sensitive measure, providing new insights into quantum state dynamics in disordered systems.

Findings

Long-time complexity scales linearly in ergodic phase

Complexity grows sublinearly in MBL phase

Eigenstate contributions differ significantly between phases

Abstract

We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic phase, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL phase, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL phase. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic phase receives contributions from almost all eigenstates, the complexity in the MBL phase is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.