The energy band theory of acoustic crystal provides an important theoretical foundation for controlling the features of sound fields. By utilizing the acoustic flat bands, we can effectively modulate the sound wave to realize the acoustic localization and diffusion. In this work, we employ an artificial gauge field to design a system supporting multiple acoustic flat bands, leading to the emergence of diversified acoustic localizations. Initially, we use cavity resonators, linked with different connectivity based on the field profiles of acoustic resonators, to emulate coupled P_z-dipole modes of atomic orbitals.
According to the band order of in-phase and out-of-phase modes in two coupled cavities, we can confirm that the cross-linked and V-shaped-linked tube structures can achieve the positive coupling and negative coupling, respectively. By introducing both positive and negative couplings in a rhombic loop, a synthetic gauge field can be formed due to the π flux phase accumulation of acoustic wave in the loop. Correspondingly, the different geometric phases of acoustic wave in different paths are analogous to the Aharonov-Bohm caging effect. Due to the Aharonov-Bohm caging effect, the introduce of π-flux in a rhombic loop causes the dispersion bands to collapse into dispersionless flat bands, providing the opportunity to control the localizations of sound fields. According to the finite structures of the cases with and without gauge fluxes, we analyze the eigenmodes and energy ratios to investigate the sound field distributions. Compared with the zero-flux structure, we find that the acoustic localization can be realized at the bulk and edge of the finite rhombic sonic crystal after introducing the artificial gauge field with π flux in each plaquette. Here the localized states, induced by Aharonov-Bohm caging effect, are topologically immune to symmetrical structure disorder, indicating that the localized mode relies on the topological feature of π-flux artificial gauge field. Additionally, based on the excitation of flat band eigenstates, the acoustic flat band bound states corresponding to different eigenstates can be obtained. By superimposing acoustic flat band bound states, we can manipulate the amplitude and phase of sound wave at specific locations, realizing the composite flat band bound states with rich acoustic field patterns. Therefore, we achieve distinct types of acoustic localized states in an acoustic topological Aharonov-Bohm cage. These localized states can be excited at any primitive cell of the rhombic lattices, and possess the remarkable ability to trap sound waves at different bulk gap frequencies, which achieves the broadband sound localizations. At the eigenfrequencies of flat bands, the localized states will transform into the extended states, exhibiting acoustic filtering functionality. Therefore, the acoustic Aharonov-Bohm cage is promising for applications at both bandgap and flat band frequencies. The findings of our study offer the theoretical guidance for exploring the acoustic localized states with artificial gauge field, and may lead to potential applications on acoustic control devices.