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However, it is only applicable to circularly\u2010polarized light and encounters substantial challenges when applied to wave fields lacking the intrinsic spin degree of freedom. Here, a new paradigm is presented for achieving geometric phase by elucidating the concept of topological complementary pair (TCP), which arises from the combination of two compact phase elements possessing opposite intrinsic topological charge. Twisting the TCP leads to the generation of a linearly\u2010varying geometric phase of arbitrary order, which is quantified by the intrinsic topological charge. Notably distinct from the conventional spin\u2010orbit coupling\u2010based theories, the proposed geometric phase is the direct result of the cyclic evolution of orbital\u2010angular\u2010momentum transformation in mode space, thereby exhibiting universality across classical wave systems. As a proof of concept, the existence of this geometric phase is experimentally demonstrated using scalar acoustic waves, showcasing the remarkable ability in the precise manipulation of acoustic waves at subwavelength scales. These findings engender a fresh understanding of wave\u2010matter interaction in compact structures and establish a promising platform for exploring geometric phase, offering significant opportunities for diverse applications in wave systems.<\/jats:p>","DOI":"10.1002\/advs.202304992","type":"journal-article","created":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T08:52:05Z","timestamp":1695372725000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Geometric Phase in Twisted Topological Complementary Pair"],"prefix":"10.1002","volume":"10","author":[{"given":"Kun","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Physics Key Laboratory of Aerospace Information Materials and Physics (MIIT) Nanjing University of Aeronautics and Astronautics (NUAA)  Nanjing 211106 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