Synthetic Magnetic Fields in Topological Acoustic Crystals: Capturing the Hofstadter Butterfly and Identifying Bulk and Edge Modes in a Coupled Resonator Lattice

Topology is fundamental in describing topological phases in acoustic metamaterials, which can be designed and fabricated using modern techniques like 3D printing. These artificial structures have introduced novel topological models, analogous to their electronic counterparts, enabling unique effects for acoustic applications. We propose computationally and experimentally test a reconfigurable 2D acoustic lattice, where the resonant frequencies are tuned via cylinders, and coupling is governed by a quantum Hamiltonian. The lattice configuration and quasi-periodicity enable the system to emulate a topological bulk-boundary correspondence, driven by nontrivial topology in virtual dimensions. By introducing a phason into the Hamiltonian, we computationally map the Hofstadter butterfly spectrum and analyze the acoustic density of states over varying frequencies. The energy spectrum reveals a fractal structure, dependent on both hopping dimerization and the strength of the on-site potential. This work has potential applications in quantum computing, where topologically protected modes provide a stable, error-resistant platform for quantum operations.