Growth of block diagonal operators and symmetry-resolved Krylov complexity

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.