From phase space to Krylov space, one shell at a time

This paper investigates the classical limit of Krylov complexity, a diagnostic tool originally developed for quantum systems to quantify operator growth and chaotic dynamics. The authors seek to determine if the Lanczos algorithm—the mathematical engine behind Krylov complexity—can be reformulated in phase space using Poisson brackets and phase-space integrals, and whether this classical construction can effectively distinguish between integrable and chaotic dynamics in semiclassical systems.

The researchers define a classical Lanczos algorithm where the Liouvillian operator is generated by the Poisson bracket. By utilizing the Stratonovich–Weyl correspondence, they prove a 'Krylov-Ehrenfest theorem,' showing that quantum Krylov quantities converge to their classical counterparts as $\hbar \to 0$. A key innovation is the introduction of a microcanonical Krylov complexity, which restricts the Lanczos recursion to specific energy shells. This allows for a fine-grained analysis of dynamics, which the authors apply to the Lipkin-Meshkov-Glick (LMG) and Feingold-Peres (FP) collective spin models.

The study demonstrates that classical Krylov complexity is a robust probe of early-time chaotic dynamics. In the LMG model, which is integrable but contains saddle points, previous quantum analyses often reported 'chaotic' signatures. The authors show that these signatures are artifacts of global spectral averaging. By using their microcanonical framework, they isolate the instability to the energy shell containing the saddle point, while recovering clear signatures of integrability (such as persistent oscillations) in all other energy regions. In contrast, the FP model exhibits uniform saturation across energy shells when chaotic, confirming that the microcanonical approach successfully differentiates between localized instabilities and genuine spectral chaos.