Universal chaotic dynamics from Krylov space

TL;DR

The paper develops a Krylov-space framework to study the time evolution of quantum states, revealing an Ehrenfest-type relation that ties Krylov complexity to the Hamiltonian spectrum. It demonstrates that early-time linear growth and late-time saturation of Krylov state complexity are not exclusive to chaotic dynamics, and shows that chaotic spectra imprint a universal rise-slope-ramp-plateau pattern in the transition probability to Krylov basis states, with a long ramp signaling spectral rigidity. A peak in Krylov complexity at late times arises from this ramp under chaotic, but not non-chaotic, spectra. The analysis, illustrated with Gaussian ensembles and SYK models, establishes Krylov-state dynamics as a sharp diagnostic of chaos with potential connections to spectral complexity and holography.

Abstract

Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.

Figures (19)

• Figure 1: The SFF (left) and Krylov complexity of maximally entangled state (right) as functions of time for the Gaussian unitary ensemble with dimension $L=1024$ and $128$ realizations. The blue curves represent the numerical results and the black lines denote the values of $1/L$ (left) and $1/2$ (right).
• Figure 2: The Lanczos coefficients of a maximally entangled state evolving with the Liouvillian $\mathcal{L}=H\otimes\mathbbm{I}$ where the $H$ is taken from the GUE with dimension $D=4096$. The black curve is a $\sqrt{1-n/D}$.
• Figure 3: The snapshots of the transition probability $\left|\phi_n(it)\right|^2$ and $\left|\phi_n(\beta)\right|^2$ on the Krylov chain in one realization of the GUE with $L=1024$.
• Figure 4: Krylov complexity as a function of inverse temperature (left) and real time (right) for the GUE with $L=1024$ and $64$ realizations. The dots represent the numerical results and the solid curves represent analytical approximation from spectral complexity.
• Figure 5: The Krylov complexities $K(t;0)$ (solid curves) of maximally entangled state and spectral complexities $C(t;0)$ (dashed curves) as functions of time $t$ in the GOE, GUE, and GSE. The Krylov complexities are calculated numerically from the $\tilde{\beta}$-ensemble with $L=256$ and $128$ realizations. The spectral complexities are derived from \ref{['SpectralIntegral']}.