Krylov-Space Memory Cores

Understanding how isolated quantum many-body systems thermalize is a cornerstone of modern physics. While generic systems follow the Eigenstate Thermalization Hypothesis (ETH), certain initial states—such as those exhibiting weak thermalization, confinement-induced dynamics, or many-body scarring—show anomalous relaxation. This paper investigates whether these atypical states share a common spatial organization within the Krylov space, a state-dependent geometric framework that maps many-body dynamics onto a one-dimensional hopping problem.

The authors introduce a stationary, depth-resolved framework to analyze the late-time behavior of quantum systems. They define the Krylov-space memory core using three complementary diagnostics:

• Residual equilibration fluctuations: Measuring the long-time variance of Krylov occupations.
• Gibbs-deviation profile: Quantifying the mismatch between the stationary occupation and a Gibbs reference state.
• Krylov-current fluctuation activity: Probing persistent probability exchange between neighboring Krylov depths.

By calculating these quantities, the researchers identify whether the late-time probability is merely present or if it carries specific, structured physical content. They compare these results across different models, including chaotic Ising chains and the PXP model, to determine if the memory core is a universal feature of anomalous dynamics.

The study demonstrates that anomalous initial states consistently develop compact memory cores near the beginning of the Krylov chain. These cores are characterized by a high concentration of residual fluctuations, Gibbs mismatch, and current-fluctuation activity. In contrast, generic thermalizing states do not exhibit this combination of signal strength and spatial compactness. The authors show that this structure is state-selective, as it appears in nonintegrable systems with anomalous dynamics but is absent in typical thermalizing states under the same Hamiltonian. Furthermore, comparisons with escaping Krylov geometries confirm that rapid spreading alone does not generate these cores.