Topological phases and phase transitions have been extensively studied in electronic [1,2], photonic , and acoustic [4,5] systems in the past decades. Recently, a new class of topological insulators, called higher-order topological insulators (HOTIs) that are characterized by higher-order bulk-boundary (e.g., bulk-corner or bulk-hinge) correspondence, were discovered [6–36]. HOTIs set up examples with multidimensional topological physics going beyond the bulk-edge correspondence in conventional topological insulators and semimetals and thus attract growing attention. Prototype HOTIs include quadrupole and octupole topological insulators [6–16,37,38], 3D HOTIs in electronic systems with topological hinge states [17–20], and HOTIs with quantized Wannier centers [21–34,39–41]. Among these prototype HOTIs, the breathing kagome lattice is regarded as an excellent platform to study higher-order topological phases and phase transitions. It was first proposed in Ref. , and subsequently experimentally realized in acoustic [22,23] and photonic [24,25] systems. In the breathing kagome lattice, the higher-order topology is characterized by the quantized bulk polarization (or the position of the Wannier center). When there is a mismatch between the Wannier center and the lattice site, the breathing kagome lattice becomes a higher-order topological phase and exhibits gapped edge states and in-gap corner states. On the contrary, the breathing kagome lattice becomes a topological trivial phase when the Wannier center overlaps with the lattice site. Despite extensive studies on HOTIs based on the breathing kagome lattice, most studies only distinguish the higher-order topological phases from the trivial phases. As a result, the distinctions between two higher-order topological phases and phase transitions have not yet been revealed.
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