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

Abstract

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 N\'eel form, and the behavior in a typical thermal state at $\ensuremath{\beta}=1$. 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 ${\ensuremath{\xi}}_{\text{OTOC}}$. For $|x|<{\ensuremath{\xi}}_{\text{OTOC}}$, information spreading occurs at the maximal velocity ${v}_{\mathrm{max}}=J$ and we confirm predictions for the early-time behavior of $C(x,t)\ensuremath{\sim}{t}^{2|x|}$. For the case of the quench starting from the N\'eel 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.