Euclidean operator growth and quantum chaos

TL;DR

This paper investigates the growth of local operators under Euclidean time evolution in lattice systems, deriving bounds that distinguish integrable from chaotic systems and connecting operator growth to quantum chaos measures.

Contribution

It establishes rigorous bounds on Euclidean operator growth, introduces an analog of Lieb-Robinson bounds for Euclidean space, and relates operator growth to chaos indicators like Lyapunov exponents.

Findings

Euclidean spatial growth is polynomial in integrable systems.

Chaotic systems exhibit exponential Euclidean growth, reaching infinity in finite time in higher dimensions.

One-dimensional systems have superexponentially suppressed power spectra at large frequencies.

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.