Fractal energy spectra arise from the interplay between two periodic parameters at different scales. Artificial lattices with nontrivial topological features in synthetic dimensions can emulate such interplay, exhibiting topological boundary correspondence and fractal structures resembling the Hofstadter butterfly. However, the realization of Hofstadter butterfly spectra in cavity magnonic lattices has received limited attention. In this work, we propose a one-dimensional cavity magnonic lattice in which each unit cell couples a cavity photon with a magnon. By tuning the system parameters to implement a synthetic magnetic flux, we demonstrate the emergence of a Hofstadter butterflylike energy spectrum and the formation of distinct edge-state modes. These edge states exhibit flipping behavior, enabling controlled storage and transmission of quantum information. We further analyze the spectral structure and compute thetopological invariants to characterize the system's phase diagram. The influence of random defects is also investigated. While small imperfections have minimal impact on topological properties, larger defects may disrupt the edge modes, highlighting the need for defect suppression to preserve topological robustness in practical implementations.

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• Circuit quantum electrodynamics
• Hybrid quantum systems
• Magnons
• Superconducting quantum optics