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A Schwinger-Keldysh Formulation of Semiclassical Operator Dynamics

Jeff Murugan, Hendrik J. R. van Zyl

TL;DR

The paper develops a real-time Schwinger–Keldysh path-integral framework for Krylov dynamics, treating Krylov complexity as an in–in observable generated on a closed time contour. It shows that the Lanczos coefficients define an emergent classical Hamiltonian on Krylov phase space, with exponential growth arising from hyperbolic trajectories when $b(n)\sim αn$, and identifies universal fixed points and perturbation classifications (irrelevant, marginal, relevant). The formalism enables controlled access to fluctuations and large deviations, revealing integrability–chaos crossovers through full counting statistics and dynamical large-deviation functions. Exact solvable models like the square-root hopping and the $SU(1,1)$ Liouvillian illustrate the mechanics of linear growth fixed points and the role of symmetry in operator growth. Overall, the work reframes Krylov complexity as a dynamical, field-theoretic problem applicable to closed and open quantum systems, uncovering universal structures and providing tools to probe fluctuations beyond mean complexity.

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.

A Schwinger-Keldysh Formulation of Semiclassical Operator Dynamics

TL;DR

The paper develops a real-time Schwinger–Keldysh path-integral framework for Krylov dynamics, treating Krylov complexity as an in–in observable generated on a closed time contour. It shows that the Lanczos coefficients define an emergent classical Hamiltonian on Krylov phase space, with exponential growth arising from hyperbolic trajectories when , and identifies universal fixed points and perturbation classifications (irrelevant, marginal, relevant). The formalism enables controlled access to fluctuations and large deviations, revealing integrability–chaos crossovers through full counting statistics and dynamical large-deviation functions. Exact solvable models like the square-root hopping and the Liouvillian illustrate the mechanics of linear growth fixed points and the role of symmetry in operator growth. Overall, the work reframes Krylov complexity as a dynamical, field-theoretic problem applicable to closed and open quantum systems, uncovering universal structures and providing tools to probe fluctuations beyond mean complexity.

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.
Paper Structure (19 sections, 116 equations, 2 figures)

This paper contains 19 sections, 116 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Krylov dynamics as a tight-binding problem.
  3. A brief review of the Schwinger-Keldysh formalism
  4. Managing expectation values
  5. Path integral representation
  6. Schwinger--Keldysh formulation of Krylov complexity.
  7. Path integral representation
  8. Examples
  9. Square-root hopping
  10. The $\mathfrak{su}(1,1)$ model
  11. Semiclassical Limit and Hyperbolic Trajectories
  12. Hyperbolic trajectories
  13. Linear growth as a universal fixed point
  14. Krylov Trajectories and Phase Transitions in Operator Growth
  15. Discussion
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Schwinger-Keldysh closed-time contour with insertions $J_+$ and $J_-$ on the forward and backward branches respectively.
  • Figure 2: Phase portrait for the semiclassical Krylov dynamics $\dot n=-2\alpha n\sin p$, $\dot p=-2\alpha\cos p$. Trajectories lie on $n\cos p=\mathrm{const}$. The vertical manifolds at $p=\pm\pi/2$ are fixed in $p$; $p=-\pi/2$ is unstable (repels in $n$) and $p=+\pi/2$ is stable (attracts in $n$).