Out-of-time-ordered correlators in quantum Ising chain

TL;DR

This paper investigates out-of-time-ordered correlators in the quantum Ising chain, revealing distinct behaviors for local and nonlocal operators and highlighting limitations in defining chaos parameters.

Contribution

It provides an exact analysis of OTOC behaviors in the quantum Ising model, contrasting local and nonlocal operator dynamics and showing limitations of chaos diagnostics.

Findings

Local operators show no scrambling, approaching original value with a t^{-1} decay.

Nonlocal operators exhibit decay to zero with a t^{-1/4} power law at criticality.

Long-time behaviors are not captured by conformal field theory and challenge chaos quantification.

Abstract

Out-of-time-ordered correlators (OTOC) have been proposed to characterize quantum chaos in generic systems. However, they can also show interesting behavior in integrable models, resembling the OTOC in chaotic systems in some aspects. Here we study the OTOC for different operators in the exactly-solvable one-dimensional quantum Ising spin chain. The OTOC for spin operators that are local in terms of the Jordan-Wigner fermions has a "shell-like" structure: after the wavefront passes, the OTOC approaches its original value in the long-time limit, showing no signature of scrambling; the approach is described by a $t^{-1}$ power law at long time $t$. On the other hand, the OTOC for spin operators that are nonlocal in the Jordan-Wigner fermions has a "ball-like" structure, with its value reaching zero in the long-time limit, looking like a signature of scrambling; the approach to zero, however, is described by a slow power law $t^{-1/4}$ for the Ising model at the critical coupling. These long-time power-law behaviors in the lattice model are not captured by conformal field theory calculations. The mixed OTOC with both local and nonlocal operators in the Jordan-Wigner fermions also has a "ball-like" structure, but the limiting values and the decay behavior appear to be nonuniversal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.