Hofstadter spectrum in a semiconductor moiré lattice

Recently, the Hofstadter spectrum of a twisted ${\mathrm{WSe}}_{2}/{\mathrm{MoSe}}_{2}$ heterobilayer was observed in experiment [C. R. Kometter et al. Nat. Phys. 19, 1861 (2023)], but the origin of Hofstadter states remains unclear. Here, we present a comprehensive theoretical interpretation of the observed Hofstadter states by calculating their accurate Hofstadter spectrum. We point out that the valley Zeeman effect, a unique feature of the transition metal dichalcogenide (TMD) materials, plays a crucial role in determining the shape of the Hofstadter spectrum, due to the narrow bandwidth of the moir\'e bands. This is distinct from the graphene-based moir\'e systems. We further predict that the Hofstadter spectrum of the moir\'e flat band, which was not observed in the experiment, can be observed in the same system with a larger twist angle ${2}^{\ensuremath{\circ}}\ensuremath{\lesssim}\ensuremath{\theta}\ensuremath{\lesssim}{3}^{\ensuremath{\circ}}$. Our theory paves the way for further studies of the interplay between the Hofstadter states and correlated insulating states in such moir\'e lattice systems.