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Energy bands of electrons in a square lattice potential threaded by a uniformmagnetic field exhibit a fractal structure known as the Hofstadter butterfly.Here we study a Fermi gas in a 2D optical lattice within a linear cavity with atilt along the cavity axis. The hopping along the cavity axis is only inducedby resonant Raman scattering of transverse pump light into a standing wavecavity mode. Choosing a suitable pump geometry allows to realize theHofstadter-Harper model with a cavity-induced dynamical synthetic magneticfield, which appears at the onset of the superradiant phase transition. Thedynamical nature of this cavity-induced synthetic magnetic field arises fromthe delicate interplay between collective superradiant scattering and theunderlying fractal band structure. Using a sixth-order expansion of the freeenergy as function of the order parameter and by numerical simulations we showthat at low magnetic fluxes the superradiant ordering phase transition is firstorder, while it becomes second order for higher flux. The dynamic nature of themagnetic field induces a non-trivial deformation of the Hofstadter butterfly inthe superradiant phase. At strong pump far above the self-ordering threshold werecover the Hofstadter butterfly one would obtain in a static magnetic field.

The Hofstadter Butterfly in a Dynamic Cavity-Induced Synthetic Magnetic Field