Temperature dependence of quantum information scrambling in gapped local systems

We study temperature dependence of quantum information scrambling, specifically, in systems with a gap. Firstly, we perform large scale tensor network based numerics in gapped chaotic one dimensional spin chains to obtain scrambling data at different temperatures. We find that our numerics work very well even at low temperatures, and we are able to determine the temperature dependence of butterfly velocity to be √(T/m), where m is the mass, for T<m. From our numerics, we also observe a broadening of the operator wavefront at finite temperatures, which had been observed in the context of infinite T previously. Secondly, we perform a perturbative calculation to study scrambling in the paramagnetic phase of a 2+1 D non-linear sigma model to analytically understand the temperature dependence of the butterfly velocity. Using the ladder diagram techniques, we verify the √(T/m) behavior of the butterfly velocity at low temperatures, T<<m. Thirdly, we discuss these results in the context of a recently proposed state dependent bound on scrambling, and show that our results are in accordance with the bound, and in fact provides a clear physical picture of scrambling at low temperatures.