Many-Body Localization of Haldane-Shastry Model with Periodic Driving

In this work, we study the property of many-body localization(MBL) in the Haldane-Shastry(HS) chain which is driven by an additional time-dependent perturbation periodically. The Haldane-Shastry (HS) model is the integrable one-dimensional quantum spin chain with long-range interactions, it is the generalized Heisenberg XXX model which only contain nearest two body interaction. By using HS model, we consider the global two-body interaction and expand the field of MBL. In this paper, we establish a Floquet operator by adding a time-periodic field formed as trigonometric function to a closed and disordered HS model in this periodic driven system. We use the exact matrix diagonalization to probe the property of MBL with different disorders and system sizes. When we drive the HS model in MBL phase, it shows that there is a significant change in the diagrams with when driving strength T reach to T-c which is the critical driving strength. We get that at large T (T > T-c), MBL phase will be broken and a transition from localized phase to delocalized phase will happen, conversely, at small T (T < T-c), MBL phase will be retained. The stronger disorder taken in system, the more stable the localized phase is and the higher T-c is needed to drive the transition. However, there is no MBL phase transition when we drive the HS model in ergodic phase with periodic driving. In contrast to the Heisenberg XXX model with the same situation which we have studied recently, the phase transition from delocalized phase to localized phase occurs. We also explore the non-disorder system of HS model with the same driving to explore the properties of MBL, it shows that under periodic driving, the non-disordered HS system has the quantum phase transition rather than MBL phase transition. This illustrates the important role of disorder on MBL.