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Inspired by the recently discovered phenomenon of re-entrant localization(REL) [Roy et al., PRL 126, 106803 (2021)], we propose a new approach to induceREL, i.e., to control the quasiperiodic potential's phase-shift between odd andeven sites, as thus the system can be dubbed as a phase-shift AAH model. Wethen analyze the participation ratios and corresponding scaling behaviors, andthe results reveal that multiple re-entrant localization (MREL) phenomenonoccurs. Furthermore, by depicting the behavior of extension dynamics, we obtaina whole visualized process of the system entering and re-entering the localizedphase multiple times. Finally, we exhibit the distribution of quasiperiodicpotential with different phase-shift and quasiperiodic parameter, and show thereason for the occurrence of MREL phenomenon, i.e., the introduction ofphase-shift enables a part of eigenstates to escape from the localized phase,thus weakening the ``localizibility'' of the system.

The multiple re-entrant localization in a phase-shift quasiperiodic chain