Growth of block diagonal operators and symmetry-resolved Krylov complexity

TL;DR

This paper introduces symmetry-resolved Krylov complexity to analyze how symmetry structures influence the growth of invariant operators over time, revealing early-time averaging and late-time sector-dependent behaviors.

Contribution

It presents the concept of symmetry-resolved Krylov complexity, providing new insights into operator growth and symmetry effects in quantum many-body systems.

Findings

Early-time complexity equals the average of symmetry-resolved contributions.

At later times, charge sectors influence complexity differently.

In some cases, Krylov complexity exhibits equipartition across sectors.

Abstract

This work addresses how the growth of invariant operators is influenced by their underlying symmetry structure. For this purpose, we introduce the symmetry-resolved Krylov complexity, which captures the time evolution of each block into which an operator, invariant under a given symmetry, can be decomposed. We find that, at early times, the complexity of the full operator is equal to the average of the symmetry-resolved contributions. At later times, however, the interplay among different charge sectors becomes more intricate. In general, the symmetry-resolved Krylov complexity depends on the charge sector, although in some cases this dependence disappears, leading to a form of Krylov complexity equipartition. Our analysis lays the groundwork for a broader application of symmetry structures in the study of Krylov space complexities with implications for thermalization and universality in many-body quantum systems.