Out of Time Ordered Correlators and Entanglement Growth 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 $\beta=1$. 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, $\xi_{OTOC}$. For $|x|<\xi_{OTOC}$ information spreading occurs at the maximal velocity $v_{max}=J$ and we confirm predictions for the early time behaviour of $C(x,t)\sim t^{2|x|}$. For the case of the quench starting from the Néel product state we also study the growth of the bipartite entanglement, focusing on the late and infinite time behaviour. The approach to a bounded entanglement is observed to be slow for the disorder strengths we study.