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We present here a Hofstadter's butterfly spectrum for the magic angle twistedbilayer graphene obtained using an ab initio based multi-million atomtight-binding model. We incorporate a hexagonal boron nitride substrate andout-of-plane atomic relaxation. The effects of a magnetic field are introducedvia the Peierls modification of the long-range tight-binding matrix elementsand the Zeeman spin splitting effects. A nanoribbon geometry is studied, andthe quantum size effects for the sample widths up to 1 $\mu$m are analyzed bothfor a large energy window and for the flatband around the Fermi level. Forsufficiently wide ribbons, where the role of the finite geometry is minimized,we obtain and plot the Hofstadter spectrum and identify the in-gap Chernnumbers by counting the total number of chiral edge states crossing these gaps.Subsequently, we examine the Wannier diagrams to identify the insulating statesat charge neutrality. We establish the presence of three types of electronicstates: moir\'e, mixed, and conventional. These states describe both the bulkLandau levels and the edge states crossing gaps in the spectrum. The evolutionof the bulk moir\'e flatband wavefunctions in the magnetic field isinvestigated, predicting a decay of the electronic density from the moir\'ecenters as the magnetic flux increases. Furthermore, the spatial properties ofthe three types of edge states are studied, illustrating the evolution of theirlocalization as a function of the nanoribbon momentum.

Atomistic theory of moir\'e Hofstadter's butterfly in magic-angle graphene