This paper investigates the temperature dependence of quantum information scrambling in local systems with an energy gap, m, above the ground state. We study the speed and shape of growing Heisenberg operators as quantified by out-of-time-order correlators, with particular attention paid to so-called contour dependence, i.e. dependence on the way operators are distributed around the thermal circle. We report large scale tensor network numerics on a gapped chaotic spin chain down to temperatures comparable to the gap which show that the speed of operator growth is strongly contour dependent. The numerics also show a characteristic broadening of the operator wavefront at finite temperature T. To study the behavior at temperatures much below the gap, we perform a perturbative calculation in the paramagnetic phase of a 2+1D O(N) non-linear sigma model, which is analytically tractable at large N. Using the ladder diagram technique, we find that operators spread at a speed T/m−−−−√ at low temperatures, T≪m. In contrast to the numerical findings of spin chain, the large N computation is insensitive to the contour dependence and does not show broadening of operator front. We discuss these results in the context of a recently proposed state-dependent bound on scrambling.

1 aSahu, Subhayan1 aSwingle, Brian uhttps://arxiv.org/abs/2005.10814