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We propose a minimal "three-patch model" for the anomalous Hall crystal (AHC), a topological electronic state that spontaneously breaks both time-reversal symmetry and continuous translation symmetry. The proposal for this state is inspired by the recently observed integer and fractional quantum Hall states in rhombohedral multilayer graphene at zero magnetic field. There, interaction effects appear to amplify the effects of a weak moir\'e potential, leading to the formation of stable, isolated Chern bands. It has been further shown that Chern bands are stabilized in mean field calculations even without a moir\'e potential, enabling a realization of the AHC state. Our model is built upon the dissection of the Brillouin zone into patches centered around high symmetry points. Within this model, the wavefunctions at high symmetry points fully determine the topology and energetics of the state. We extract two quantum geometrical phases of the non-interacting wavefunctions that control the stability of the topologically nontrivial AHC state. The model predicts that the AHC state wins over the topological trivial Wigner crystal in a wide range of parameters, and agrees very well with the results of full self-consistent Hartree-Fock calculations of the rhombohedral multilayer graphene Hamiltonian.
Soejima et al. (Fri,) studied this question.