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Theory of electron states in a twisted two-valley 2D system

A system similar to gapped graphene (for example, fluorinated) containing two or more electron valleys is considered. It is assumed that the material has a sector cut and is deformed in the plane and the the cut edges are connected to form an adiabatically curved atomic net without extended defects. We neglect the deformation potential. In such a system, the local momentum of the valley center ${\bf K}$ acts as the vector potential of fictitious magnetic field. We found the electron states in such system in the case of orientation ${\bf K}$ along the azimuth of geometric space at any point. It is shown that the vector potential results in the appearance of local discrete electron states. Mathematically, the problem is mapped onto the Coulomb problem with an effective charge depending on ${\bf K}$.