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

TL;DR

The paper probes maximal quantum chaos through an effective field theory (EFT) that enforces a maximal Lyapunov exponent $λ_L=2πT$ via a shift symmetry acting on the hydrodynamic chaos mode. It connects OTOCs to Krylov complexity by computing Lanczos coefficients and Krylov exponents $λ_K$ for correlators that satisfy shift symmetry, KMS, unitarity, and reality constraints. The main finding is that $λ_K$ can take both $λ_L$ and $λ_L/2$, demonstrating that shift symmetry alone does not guarantee maximal Krylov growth and highlighting tension with the conjectured bound $λ_L≤λ_K≤2πT$. The work also shows that some autocorrelator solutions yield power spectra resembling the thermal product formula seen in holography, while others reveal differing analytic structures, including higher-order poles and zeros, underscoring the nuanced relationship between IR EFT dynamics and UV-sensitive spectral data.

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, $λ_L = 2πT$, 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 $λ_K = λ_L$ and $λ_K = λ_L/2$. 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 $λ_L \leq λ_K \leq 2πT$. Finally, we identify autocorrelator solutions whose power spectra closely resemble the so-called thermal product formula seen in holographic systems.

Figures (1)

• Figure 1: Two perspectives on operator growth. Left: spread of the operator cloud in the space of degrees of freedom, driven by the hydro mode $\epsilon$. Right: spreading across the Krylov basis in Hilbert space, governed by the Lanczos sequence $\{b_n\}$. We will identify the initial operator $\mathcal{O}(0)$ with the bare field operator in the EFT.