Quantum chaos and late-time equipartition of symmetry-resolved Krylov complexity

Understanding quantum chaos in many-body systems requires robust diagnostics. While traditional measures like level-spacing distributions have long been used, Krylov complexity has recently emerged as a powerful tool to quantify operator growth under Heisenberg evolution. In systems with conserved charges, symmetries organize the dynamics into distinct sectors. Although early-time relations between full and symmetry-resolved Krylov complexities are known, their behavior at late-time saturation has remained unclear. This paper investigates whether symmetry-resolved Krylov complexity exhibits a clean sector-wise structure at late times in finite-dimensional chaotic quantum systems.

The authors analyze the operator dynamics of finite-dimensional chaotic systems possessing a conserved charge. By decomposing both the Hamiltonian and a symmetry-invariant seed operator into blocks corresponding to distinct symmetry sectors, they formulate independent Krylov chains within each sector. Using the spectral formulation of operator Krylov dynamics, they evaluate the dimensions of the accessible Krylov spaces. By linking this to the late-time delocalization of the Krylov wavefunctions across the accessible Krylov space, they determine the saturation value of both the full and the symmetry-resolved complexities.

The analysis reveals that, after saturation, the unresolved Krylov complexity becomes strictly additive over the symmetry sectors. In the absence of accidental Liouvillian degeneracies, the fractional complexity of a given sector is controlled by the dimension of its accessible operator space, leading to a dimension-weighted equipartition. Unlike early-time behavior, which retains explicit dependence on initial operator projections, the late-time structure is governed solely by the dimensions of the invariant subspaces. Consequently, symmetry resolution is crucial for correctly interpreting the saturation plateau of Krylov complexity as a signature of chaotic operator growth.