Coalescence of multiple topological orders in quasi-one-dimensional bismuth halide chains.

Topology is being widely adopted to understand and to categorize quantum matter in modern physics. The nexus of topology orders, which engenders distinct quantum phases with benefits to both fundamental research and practical applications for future quantum devices, can be driven by topological phase transition through modulating intrinsic or extrinsic ordering parameters. The conjoined topology, however, is still elusive in experiments due to the lack of suitable material platforms. Here we use scanning tunneling microscopy, angle-resolved photoemission spectroscopy, and theoretical calculations to investigate the doping-driven band structure evolution of a quasi-one-dimensional material system, bismuth halide, which contains rare multiple band inversions in two time-reversal-invariant momenta. According to the unique bulk-boundary correspondence in topological matter, we unveil a composite topological phase, the coexistence of a strong topological phase and a high-order topological phase, evoked by the band inversion associated with topological phase transition in this system. Moreover, we reveal multiple-stage topological phase transitions by varying the halide element ratio: from high-order topology to weak topology, the unusual dual topology, and trivial/weak topology subsequently. Our results not only realize an ideal material platform with composite topology, but also provide an insightful pathway to establish abundant topological phases in the framework of band inversion theory.