Out-of-time ordered correlators and entanglement growth in the random-field XX spin chain

We study out-of-time ordered 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 nonzero 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 behavior 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 ξOTOC. For |x|<ξOTOC, information spreading occurs at the maximal velocity vmax=J and we confirm predictions for the early-time behavior of C(x,t)∼t2|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 behavior. The approach to a bounded entanglement is observed to be slow for the disorder strengths we study.