Krylov Complexity as a Probe for Chaos

TL;DR

The paper investigates Krylov complexity as a diagnostic of quantum chaos in many-body systems. It develops analytic expressions for Krylov evolution via Lanczos coefficients and analyzes late-time saturation. It shows that chaotic and integrable dynamics share growth and saturation, but chaotic systems reach the infinite-time average at saturation time and may exhibit a peak; integrable systems approach from below over longer times, with initial-state and level-spacing statistics governing the behavior. Numerical tests on a spin-1/2 Ising chain corroborate the analytic predictions, highlighting the role of the inverse participation ratio and special initial states in revealing a peak. The work connects Krylov complexity to thermalization and spectral complexity, suggesting Krylov complexity as a practical chaos probe beyond OTOCs.

Abstract

In this work, we explore in detail, the time evolution of Krylov complexity. We demonstrate, through analytical computations, that in finite many-body systems, while ramp and plateau are two generic features of Krylov complexity, the manner in which complexity saturates reveals the chaotic nature of the system. In particular, we show that the dynamics towards saturation precisely distinguish between chaotic and integrable systems. For chaotic models, the saturation value of complexity reaches its infinite time average at a finite saturation time. In this case, depending on the initial state, it may also exhibit a peak before saturation. In contrast, in integrable models, complexity approaches the infinite time average value from below at a much longer timescale. We confirm this distinction using numerical results for specific spin models.

Figures (5)

• Figure 1: Time evolution of complexity for different initial states $|X+\rangle, |Y+\rangle$ and $|Z+\rangle$ for the case where the model is integrable ($h=0$) and chaotic ($h=0.5$). The computation is performed for $N=11$.
• Figure 2: Time evolution of complexity for initial state given by \ref{['ISYZ']} for the case where the model is integrable ($h=0$) and chaotic ($h=0.5$). The computation is performed for $N=11$.
• Figure 3: Time evolution of complexity for initial state given by \ref{['TS']} with $\beta=0$. One observes that in the chaotic case the complexity exhibits a clear peak before saturation. The computation is performed for $N=11$.
• Figure 4: Time evolution of complexity for initial state $|Y+\rangle$. for $h=1,1.5,2, 3$. This shows that as we progress towards higher values of $h$, the system moves away from chaos and leans towards integrability. The computation is performed for $N=11$.
• Figure 5: Level spacing for energy eigenvalues of the model \ref{['Ising']} with positive parity for different values of $h$ given by $h=1,1.5,2, 3$. This shows that as we progress towards higher values of $h$, the distribution moves away from Wigner and leans towards Poisson. To have a better statistic the computation is performed for $N=13$.