Operator Krylov complexity in random matrix theory

TL;DR

This paper investigates Krylov complexity within Random Matrix Theory, revealing universal behaviors, bounds in chaotic systems, and temperature-dependent growth patterns, thereby connecting operator complexity with quantum chaos and thermal effects.

Contribution

It analytically characterizes Krylov complexity growth in RMT at different temperatures and proposes bounds for chaotic systems based on RMT behavior.

Findings

At infinite temperature, Krylov complexity grows linearly with time.

The Lanczos coefficient $b_n$ saturates to a constant plateau in large $N$ limit.

At low temperature, complexity exhibits exponential growth before scrambling, then linear growth afterward.

Abstract

Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity $\mathcal{K}(t)$ in Random Matrix Theory (RMT). In large $N$ limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient $\{b_n\}$ saturate to constant plateau $\lim\limits_{n\rightarrow\infty}b_n=b$, rendering a linear growing complexity $\mathcal{K}(t)\sim t$, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value $b$ to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the $\{b_n\}$ in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that $b_n$ will first exhibit linear growth with $n$, whose slope saturates the famous chaos bound. After hitting the same plateau $b$, $b_n$ will then remain constant. This indicates $\mathcal{K}(t)\sim e^{2\pi t/\beta}$ before scrambling time $t_*\sim O(\beta\log\beta)$, and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite $N$ corrections.