Nonlinear Berry Phase In The One-dimensional Periodic Quantum Wells

BEC (Bose-Einstein Condensate) in the periodic potential wells is one of the hot research problems on the field of ultracold atoms. As one typical model, the Bose-Hubbard model has been an important model to investigate the quantum phase transition and other contents of condensed matter physics. Berry phase is another important concept in the the condensed matter physics and plays an important role in various respects in condensed matter research. In this thesis, starting from an exact nonlinear Bloch solutions, and with the help of its Berry phase, we found the correct method to construct the corresponding Wannier function, by which the Hamiltonian can be maped from real space to the occupation number representation.One exact formula for Berry phase in Condensed Matter Physics periodical potential is presented, based Condensed Matter Physics on the exact Bloch solutions for Nonlinear Kronig-Penny model. And the correct Wannier functions are constructed from those Condensed Matter Physics Bloch solutions, with the help of Berry phase. We found the Berry phase depends on the quais-momentum k, and the effects of the nonlinear interaction on the berry phase perform a periodic behavior with quasi momentum k. A significant effect can be found around (n+12)π/T, while nothing effect nπ/T.