Magnetic Hofstadter cascade in a twisted semiconductor homobilayer

Transition metal dichalcogenide moir\'e homobilayers have emerged as a platform in which magnetism, strong correlations, and topology are intertwined. In a large magnetic field, the energetic alignment of states with different spin in these systems is dictated by both strong Zeeman splitting and the structure of the Hofstadter's butterfly spectrum, yet the latter has been difficult to probe experimentally. Here we conduct local thermodynamic measurements of twisted WSe$_2$ homobilayers that reveal a cascade of magnetic phase transitions. We understand these transitions as the filling of individual Hofstadter subbands, allowing us to extract the structure and connectivity of the Hofstadter spectrum of a single spin. The onset of magnetic transitions is independent of twist angle, indicating that the exchange interactions of the component layers are only weakly modified by the moir\'e potential. In contrast, the magnetic transitions are associated with stark changes in the insulating states at commensurate filling. Our work achieves a spin-resolved measurement of Hofstadter's butterfly despite overlapping states, and it disentangles the role of material and moir\'e effects on the nature of the correlated ground states.