A Schwinger-Keldysh Formulation of Semiclassical Operator Dynamics

TL;DR

This paper develops a real-time Schwinger-Keldysh approach to Krylov dynamics, revealing phase-space structures, chaos signatures, and fluctuation diagnostics in operator growth within quantum systems.

Contribution

It introduces a novel Schwinger-Keldysh formulation for Krylov complexity, connecting operator dynamics to phase-space and field-theoretic descriptions, and enables analysis of fluctuations and chaos crossovers.

Findings

Exponential complexity growth linked to hyperbolic trajectories.

Universal linear Lanczos growth as a chaotic fixed point.

Controlled access to fluctuations and chaos signatures.

Abstract

In this work we develop a real-time Schwinger-Keldysh formulation of Krylov dynamics that treats Krylov complexity as an in-in observable generated by a closed time contour path integral. The resulting generating functional exposes an emergent phase-space description in which the Lanczos coefficients define an effective Hamiltonian governing operator motion along the Krylov chain. In the semiclassical limit, exponential complexity growth arises from hyperbolic trajectories, and asymptotically linear Lanczos growth appears as a universal chaotic fixed point, with sub-leading deformations classified as irrelevant, marginal or relevant. Going beyond the saddle, the Schwinger-Keldysh framework provides controlled access to fluctuations and large deviations of Krylov complexity, revealing sharp signatures of integrability-chaos crossovers that are invisible at the level of the mean. This formulation reorganises Krylov complexity into a dynamical field-theoretic framework and identifies new fluctuation diagnostics of operator growth in closed quantum systems.