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

• URL: http://arxiv.org/abs/2603.25724v1
• Date: Thu, 26 Mar 2026 17:58:18 GMT
• Title: Krylov-space anatomy and spread complexity of a disordered quantum spin chain
• Abstract summary: We show that the long-time Krylov spread complexity distinguishes the two phases.<n>In the ergodic phase, the infinite-time complexity scales linearly with the Fock-space dimension.<n>By contrast, it grows sublinearly in the MBL phase, implying that the long-time state occupies only a vanishing fraction of the chain.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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