Krylov complexity and chaos in quantum mechanics

TL;DR

This work considers the stadium billiard as a typical example of the quantum mechanical system obtained by quantizing a classically chaotic system, and numerically evaluates Krylov complexity for operators and states, finding a clear correlation between variances of Lanczos coefficients and classical Lyapunov exponents, and also a correlation with the statistical distribution of adjacent spacings of the Quantum energy levels.

Abstract

Recently, Krylov complexity was proposed as a measure of complexity and chaoticity of quantum systems. We consider the stadium billiard as a typical example of the quantum mechanical system obtained by quantizing a classically chaotic system, and numerically evaluate Krylov complexity for operators and states. Despite no exponential growth of the Krylov complexity, we find a clear correlation between variances of Lanczos coefficients and classical Lyapunov exponents, and also a correlation with the statistical distribution of adjacent spacings of the quantum energy levels. This shows that the variances of Lanczos coefficients can be a measure of quantum chaos. The universality of the result is supported by our similar analysis of Sinai billiards. Our work provides a firm bridge between Krylov complexity and classical/quantum chaos.

Figures (35)

• Figure 1: Geometry of the stadium billiard. Dirichlet boundary conditions are imposed on the boundaries.
• Figure 2: The $a/R$ dependence of the Lyapunov exponent and the ratio.
• Figure 3: The Lanczos coefficients for the truncated momentum operator $P$ in stadium billiards with $a/R=0$ (blue dots) and $a/R=1$ (orange dots). Note that the horizontal axis is in log scale. The inset is the enlarged version, where the data are used to calculate the variance.
• Figure 4: The time dependence of Krylov operator complexity for various values of $a/R$.
• Figure 5: The $a/R$ dependence of the late-time value of Krylov operator complexity.