Out of Time Ordered Correlators in the Random Field XX Spin Chain*

We study out of time order correlations, C(x,t) and entanglement growth in the random field XX model with open boundary conditions using the exact Jordan-Wigner transformation to a fermionic Hamiltonian. For any non-zero strength of the random field this model describes an Anderson insulator. Two scenarios are considered: A global quench with the initial state corresponding to a product state of the Néel form, and the behaviour in a typical thermal state at β=1. As a result of the presence of disorder the information spreading as described by the out of time correlations stops beyond a typical length scale,
ξ_OTOC. For |x|<ξ_OTOC information spreading occurs at the maximal velocity vmax=J and we confirm predictions for the early time behaviour of C(x,t)∼t^2|x|.