Euclidean operator growth and quantum chaos

TL;DR

The paper develops rigorous bounds on Euclidean-time evolution of local operators in lattice systems, revealing stark dimensional differences: 1D systems exhibit at most double-exponential norm growth and exponential spatial spreading, while higher dimensions can show finite-β* divergences and potentially instantaneous-like spreading. Central to the analysis is counting lattice-animal histories to bound nested commutators, with exact results on Bethe lattices and conjectured tight bounds in higher dimensions, and a Euclidean Lieb-Robinson bound for spatial growth. The authors connect these operator-growth bounds to matrix elements, power spectra, and Lanczos coefficients via a Dyck-path path-integral framework, deriving an improved chaos bound linked to the analytic structure of correlation functions. They illustrate how Euclidean growth may bridge ETH and OTOC-based notions of chaos, offering insights into Krylov-space delocalization and providing guidance for numerical Euclidean methods and the understanding of chaos in many-body quantum systems.

Abstract

We consider growth of local operators under Euclidean time evolution in lattice systems with local interactions. We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth. In contrast to the Minkowski case when ballistic spreading of operators is universal, in the Euclidean case spatial growth is system-dependent and indicates if the system is integrable or chaotic. In the integrable case, the Euclidean spatial growth is at most polynomial. In the chaotic case, it is the fastest possible: exponential in 1D, while in higher dimensions and on Bethe lattices local operators can reach spatial infinity in finite Euclidean time. We use bounds on the Euclidean growth to establish constraints on individual matrix elements and operator power spectrum. We show that one-dimensional systems are special with the power spectrum always being superexponentially suppressed at large frequencies. Finally, we relate the bound on the Euclidean growth to the bound on the growth of Lanczos coefficients. To that end, we develop a path integral formalism for the weighted Dyck paths and evaluate it using saddle point approximation. Using a conjectural connection between the growth of the Lanczos coefficients and the Lyapunov exponent controlling the growth of OTOCs, we propose an improved bound on chaos valid at all temperatures.

Figures (3)

• Figure 1: One-dimensional lattice with short-range interactions Hamiltonian $H=\sum_{I=1}^5 h_I$. Local operator $A$ sits at a third site counting from the left, between second and third bonds. Bonds highlited in gray form a lattice animal $I=2,3,4$.
• Figure 2: Lyapunov exponent $\lambda_{\rm OTOC}$ for the SYK model as a function of parameter $v$, which is related to temperature, $\pi v T=\cos(\pi v/2)$. Limit $v\rightarrow 0$ corresponds to high temperatures, $v\rightarrow 1$ to small temperatures. Blue line -- exact analytic result \ref{['OTOCexact']}, orange dashed line -- improved bound \ref{['OTOC']} with $2\beta^*=1$, green dotted line -- original Maldacena-Shenker-Stanford bound $2\pi T$.
• Figure 3: Lanczos coefficients $b_n^2$ associated with the moments $M_{2k}=B_{2k}\equiv B_{2k}(1)$. Choosing different $m$ in \ref{['mock']} leads to a qualitatively similar behavior.