OTOC and Quamtum Chaos of Interacting Scalar Fields

TL;DR

The paper investigates quantum chaos in interacting scalar fields by studying the thermal OTOC in a lattice-regularized $\lambda\phi^4$ theory. It adopts Hashimoto's second-quantization perturbative framework to derive analytic expressions for the spectrum, states, and coordinate matrix elements up to second order, then numerically evaluates the thermal OTOC $C_T(t)$. The results show an early-time exponential growth $C_T(t) \sim e^{2\lambda t}$ with a Lyapunov exponent scaling $\lambda \sim T^{1/4}$, and extend the chaos analysis to closed chains of $N=3,4$ oscillators, arguing that the same chaotic features persist in the 1+1D interacting scalar field theory. Overall, the work demonstrates that perturbative OTOC computations in a lattice scalar field model capture quantum-chaotic dynamics and establish a scalable method for exploring chaos in low-dimensional quantum field theories.

Abstract

Discretizing the $λφ^4$ scalar field theory on a lattice yields a system of coupled anharmonic oscillators with quadratic and quartic potentials. We begin by analyzing the two coupled oscillators in the second quantization method to derive several analytic relations to the second-order perturbation, which are then employed to numerically calculate the thermal out-of-time-order correlator (OTOC), $C_T(t)$. We find that the function $C_T(t)$ exhibits exponential growth over a long time window in the early stages, with Lyapunov exponent $λ\sim T^{1/4}$, which diagnoses quantum chaos. We furthermore investigate the quantum chaos properties in a closed chain of N coupled anharmonic oscillators, which relates to the 1+1 dimensional interacting quantum scalar field theory.