Year: 2026
Type: article
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 regime, 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 regime, 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 regime. 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 regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.
DOI: 10.1103/3xnt-2n8v
Authors: Bikram Pain, David E. Logan, Sthitadhi Roy
Institutions: Tata Institute of Fundamental Research, University of Oxford
Language: English
Topic: Quantum many-body systems
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: G1021662744, G2487168033, G8724963187
OpenAlex ID: W7163041982