Integrability breaking in semiclassical strings in Koopman-Krylov space

Abstract

While very powerful, integrability in semiclassical string solutions is known to be a rare property. Motivated by the need to understand and characterise the large landscape of non-integrable string dynamics, we extend Krylov methods for probing chaos to classical systems. We introduce a Koopman-Krylov framework, formulated in the Koopman-von Neumann description of classical mechanics and implemented via a generator extended dynamic mode decomposition (gEDMD) approximation of the Koopman generator acting on observables. Using this framework, we study how integrability-breaking deformations of integrable string dynamics induce characteristic redistributions of spectral weight, leading to observable-dependent delocalisation and spreading in Krylov space. We illustrate the Koopman-Krylov diagnostics across three classes of non-integrable semiclassical string solutions.

Figures (17)

• Figure 1: Schematic illustration of the Koopman picture. A nonlinear trajectory generated by a classical flow $F$ on phase space (left) is lifted to linear evolution of observables under the Koopman operator $U_F$ (right), which acts on functions rather than points. Different observables evolve linearly in an common Hilbert space, encoding the same underlying nonlinear dynamics.
• Figure 2: Schematic observable spectral measures under Hamiltonian flow. Integrable dynamics yields pure point measures. Weak non-integrability produces mixed measures, whose finite-resolution signature is "packetisation" of selected spectral lines. Fully mixing dynamics corresponds to purely continuous measures.
• Figure 3: Schematic overview of the numerical pipeline: dynamical sampling of trajectories, gEDMD approximation of the Koopman generator in a finite dictionary, and Krylov-based evolution used to extract spectral diagnostics.
• Figure 4: Schematic illustration of how weak integrability breaking introduces, in classes of system as the higher-derivative $su(2)$-sector, additional dynamical directions and, hence, resonant channels. Left: in the integrable truncation, motion is confined to invariant action-angle tori and observables decompose into discrete quasi-periodic frequencies. Right: the deformation activates additional degrees of freedom, allowing slow, resonance-driven coupling between otherwise decoupled sectors.
• Figure 5: Left: Krylov complexity $C_K(\sigma)$ for two representative observables, the sector-resolved energy $g_E$ and the momentum-type probe $g_P$, comparing the integrable truncation (Int, in dashed lines) with the two-loop deformed system (Def, in solid lines) at fixed deformation point $(a_1,a_2)= (0.2,-0.6)$. Right: small-$\sigma$ zoom highlighting the onset of delocalisation. Shaded bands indicate the empirical uncertainty from the ensemble of initial conditions.