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We develop an effective extended Hubbard model to describe the low-energyelectronic properties of the twisted bilayer graphene. By using the Blochstates in the effective continuum model and with the aid of the maximallylocalized algorithm, we construct the Wannier orbitals and obtain an effectivetight-binding model on the emergent honeycomb lattice. We found the Wannierstate takes a peculiar three-peak form in which the amplitude maxima arelocated at the triangle corners surrounding the center. We estimate the directCoulomb interaction and the exchange interaction between the Wannier states. Atthe filling of two electrons per super cell, in particular, we find anunexpected coincidence in the direct Coulomb energy between a charge-orderedstate and a homogeneous state, which would possibly lead to an unconventionalmany-body state.

Maximally-localized Wannier orbitals and the extended Hubbard model for the twisted bilayer graphene