Numerics of Fast Scrambling in the SYK Model

Out-of-time-order correlators (OTOCs) offer a useful tool for characterizing the approach to chaos in strongly coupled quantum systems. However, exact diagonalization numerics of OTOCs are limited to small system sizes and controlled finite size extrapolation techniques are few. To this end, we utilize massively parallel Krylov methods to calculate OTOCs in the Sachdev-Ye-Kitaev (SYK) model for more than 50 Majorana fermions. For small systems, we show that, consistent with prior results, the Lyapunov exponent exhibits a temperature dependence opposite to that of the analytic expectation. However, for large enough systems, we find that this behavior flips and the correct qualitative trend emerges. More quantitatively, we develop a finite-size rescaling procedure for extracting Lyapunov exponents and compare our results to exact calculations in the thermodynamic limit. We observe excellent agreement at high temperatures and systematic improvements in the low-temperature behavior as we scale to larger system sizes.