Out-of-time-ordered Correlators in Integrable Quantum Ising Model

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. In this work, we study the OTOC in the 1d quantum Ising model. We find that 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 starting value in the long-time limit in a $t^{-1}$ fashion, showing no sign of scrambling. On the other hand, the OTOC for spin operators that are non-local 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 $t^{-1/4}$. 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 non-local operators in the Jordan-Wigner fermions also has a ``ball-like'' structure, but the limiting values and the decay behavior appear to be non-universal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.