Krylov exponents and power spectra for maximal quantum chaos: an EFT approach

TL;DR

This paper explores the relationship between quantum chaos measures, Krylov complexity, and effective field theory, revealing that shift symmetry alone does not guarantee maximal Krylov exponents and analyzing their spectral properties.

Contribution

It demonstrates that shift symmetry in EFT constrains autocorrelators but does not enforce maximal Krylov exponents, challenging previous conjectures about chaos bounds.

Findings

Krylov and Lyapunov exponents can differ by a factor of two.

Shift symmetry constrains autocorrelators but not Krylov exponents.

Power spectra of autocorrelators resemble holographic thermal formulas.

Abstract

We examine the effective field theory (EFT) of maximal chaos through the lens of Krylov complexity and the Universal Operator Growth Hypothesis. We test the relationship between two measures of quantum chaos: out-of-time-ordered correlators (OTOCs) and Krylov complexity. In the EFT, a shift symmetry of the hydrodynamic modes enforces the maximal Lyapunov exponent in OTOCs, $\lambda_L = 2\pi T$, while simultaneously constraining thermal two-point autocorrelators. We solve these constraints on the autocorrelator, and calculate the Lanczos coefficients and Krylov exponents for several examples, finding both $\lambda_K = \lambda_L$ and $\lambda_K = \lambda_L/2$. This demonstrates that, within the EFT, the shift symmetry alone is insufficient to enforce maximal Krylov exponents even when the Lyapunov exponent is maximal. In particular, this result suggests a tension with the conjectured bound $\lambda_L \leq \lambda_K \leq 2\pi T$. Finally, we identify autocorrelator solutions whose power spectra closely resemble the so-called thermal product formula seen in holographic systems.