Finite-Size Scaling of Out-of-Time-Ordered Correlators at Late Times

Chaotic dynamics in quantum many-body systems scrambles local information so that at late times it can no longer be accessed locally. This is reflected quantitatively in the out-of-time-ordered correlator of local operators, which is expected to decay to 0 with time. However, for systems of finite size, out-of-time-ordered correlators do not decay exactly to 0 and in this paper we show that the residual value can provide useful insights into the chaotic dynamics. When energy is conserved, the late-time saturation value of the out-of-time-ordered correlator of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We provide both analytical arguments and numerical simulations to support this conclusion.

Figure 1

Top panel: Finite-size scaling of late-time OTOC $F_n^x$ (blue), $F_n^z$ (red) for $n=5,6,\dots ,15$. The lines are power-law fits $0.449n^{-0.797}$ (blue), $0.254n^{-1.047}$ (red) to the last few data points. Bottom panel: Finite-size scaling of the errors $|F_n^x-G_n^x|$ (blue), $|F_n^z-G_n^z|$ (red) for $n=5,6,\dots ,15$. The lines are power-law fits $1.998n^{-1.946}$ (blue), $0.117n^{-1.851}$ (red) to the last few data points.