Cosmological Krylov Complexity

TL;DR

This work studies operator growth and chaos in cosmological perturbations by computing Krylov complexity in a de Sitter background using a two-mode squeezing description with an effective sound speed $c_s$. The authors construct the Krylov basis via the Lanczos algorithm, obtain linear Lanczos coefficients $b_n$ and show the Krylov complexity satisfies $K=\sinh^2 r_k$ with a Lyapunov exponent $\lambda=2\alpha=|\beta_k|$, indicating chaotic dynamics. In the de Sitter case with $c_s=1$, they derive exact squeezing parameters and closed-form expressions $K=\frac{1}{4k^2\tau^2}$, $b_n=\frac{n}{2\tau}$, and $\lambda=\frac{1}{\tau}$; early times yield negligible complexity, while late times exhibit exponential growth, reflecting chaos driven by the expanding background. For $0.024\le c_s\le 1$, $b_n$ and $\lambda$ acquire explicit mode- and time-dependence, obtained numerically, and Krylov complexity is contrasted with entanglement entropy, showing substantial differences at large squeezing and supporting Krylov complexity as a meaningful probe of cosmological quantum dynamics beyond entanglement saturation.

Abstract

In this paper, we study the Krylov complexity ($K$) from the planar/inflationary patch of the de Sitter space using the two mode squeezed state formalism in the presence of an effective field having sound speed $c_s$. From our analysis, we obtain the explicit behavior of Krylov complexity ($K$) and lancoz coefficients ($b_n$) with respect to the conformal time scale and scale factor in the presence of effective sound speed $c_s$. Since lancoz coefficients ($b_n$) grow linearly with integer $n$, this suggests that universe acts like a chaotic system during this period. We also obtain the corresponding Lyapunov exponent $λ$ in presence of effective sound speed $c_s$. We show that the Krylov complexity ($K$) for this system is equal to average particle numbers suggesting it's relation to the volume. Finally, we give a comparison of Krylov complexity ($K$) with entanglement entropy (Von-Neumann) where we found that there is a large difference between Krylov complexity ($K$) and entanglement entropy for large values of squeezing amplitude. This suggests that Krylov complexity ($K$) can be a significant probe for studying the dynamics of the cosmological system even after the saturation of entanglement entropy.

Figures (3)

• Figure 1: Krylov complexity as a function of conformal time $\tau$ for exponentially expanding de Sitter universe with different wave numbers $k$. Krylov complexity grows exponentially with $\tau$ which is a sign of chaotic system
• Figure 2: Lyapunov exponent as function of conformal time $\tau$ for different values of $c_s$ and k. For each color, solid line belongs to $k = 1$, dashed line to $k = 0.1$ and dashed medium to $k = 0.01$.
• Figure 3: Krylov complexity as function of scale factor $a$ for different values of $c_s$