Hofstadter Butterfly Variants in Inhomogeneous Magnetic Fields: Quantum Simulation with Superconducting Circuits

ABSTRACT Quantum simulation offers an effective approach to explore novel phenomena in complex quantum systems, such as the famous Hofstadter butterfly that are inaccessible in conventional natural systems. We investigate variants of the Hofstadter butterfly arising in lattices subject to inhomogeneous magnetic fields. Using a superconducting‐circuit platform, we implement a zigzag spin chain and a two‐leg spin ladder, where spatially modulated interaction phases produce position‐dependent magnetic fluxes through the plaquettes. By periodically modulating the resonance frequency of superconducting qubits, we engineer the desired synthetic magnetic field. The resulting single‐excitation dynamics map onto a tight‐binding model of a charged particle moving in a non‐uniform magnetic field, enabling the exploration of fractal energy spectra beyond the uniform‐field case. We numerically simulate the system dynamics using experimentally realistic parameters and reveal variants of the Hofstadter butterfly through a spectroscopic method. The simulation results demonstrate a high degree of agreement with theoretical predictions. Our proposal provides a promising route for investigating exotic fractal energy spectra via quantum simulation.