We study the dynamics of microscopic quantum correlations, viz., bipartite entanglement and quantum discord between nearest neighbor sites, in Ising spin chain with a periodically varying external magnetic field along the transverse direction. Quantum correlations exhibit periodic revivals with the driving cycles in the finite-size chain. The time of first revival is proportional to the system size and is inversely proportional to the maximum group velocity of Floquet quasi-particles. On the other hand, the local quantum correlations in the infinite chain may get saturated to non-zero values after a sufficiently large number of driving cycles. Moreover, we investigate the convergence of local density matrices, from which the quantum correlations under study originate, towards the final steady-state density matrices as a function of driving cycles. We find that the geometric distance, $d$, between the reduced density matrices of non-equilibrium state and steady-state obeys a power-law scaling of the form $d sim n^{-B}$, where $n$ is the number of driving cycles and $B$ is the scaling exponent. The steady-state quantum correlations are studied as a function of time period of the driving field and are marked by the presence of prominent peaks in frequency domain. The steady-state features can be further understood by probing band structures of Floquet Hamiltonian and purity of the bipartite state between nearest neighbor sites. Finally, we compare the steady-state values of the local quantum correlations with that of the canonical Gibbs ensemble and infer about their canonical ergodic properties. Moreover, we identify generic features in the ergodic properties depending upon the quantum phases of the initial state and the pathway of repeated driving that may be within the same quantum phase or across two different equilibrium phases.
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