Periodically driven integrable systems with long-range pair potentials

We study periodically driven closed systems with a long-ranged Hamiltonian by considering a generalized Kitaev chain with pairing terms which decay with distance as a power law characterized by exponent alpha. Starting from an initial unentangled state, we show that all local quantities synchronize with the driving frequency omega at late times and relax to well-defined steady state values in the thermodynamic limit and after n >> 1 drive cycles for any alpha and omega. We introduce a distance measure, D-l(n), that characterizes the approach of the reduced density matrix of a subsystem of l sites to that of the final steady state. We chart out the n dependence of D-l(n) and identify a critical value alpha = alpha(c) (which depends only on the tune-averaged Hamiltonian) below which they generically decay to zero as (omega/n)(1/2). For alpha > alpha(c), in contrast, D-l(n) (omega/n)(3/2) [(omega/n)(1/2)] for omega -> infinity [0] with at least one intermediate dynamical transition. An identical behavior is found for relaxation of all nontrivial correlation functions to their steady-state values. We also study the mutual information propagation to understand the nature of the entanglement spreading in space with increasing n for such periodically driven long-ranged systems. We point out existence of qualitatively new features (absent for omega >> 1) in the space-time dependence of mutual information for omega < omega c((1)) where omega c((1)) is the largest critical frequency for the dynamical transition for a given alpha such as the presence of multiple light cone-like structures which persists even when alpha is large. We also show that the space-time dependence of the mutual information of long-ranged Hamiltonians with alpha < 2 differs qualitatively from those with alpha > 2 for any drive frequency and relate this to the behavior of the Floquet group velocity of such driven system.