Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the “ordered” state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as $\log (t)$ in the two “phases,” while at the critical point it grows as ${\log }^{\alpha }(t)$, with $\alpha$ a universal number. This universal entanglement growth requires generic (“integrability breaking”) interactions to be added to the pure transverse field Ising model.

• Received 14 August 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.217204