| To study condensation, fragmentation, fermionization and crystallization of the trapped bosonic system we consider the quantum many-body state as a linear combination of general symmetrized Hartree-Products (permanents). The optimal expansion coefficients for the ground and excited states are obtained by diagonalizing respective secular Hamiltonian. In this work we demonstrate that for any practical computation where the configurational space is restricted, the description of trapped bosonic systems strongly depends on the choice of the many-body basis set used, i.e., self-consistency is of great relevance. We develop general and complete variational many-body theory where the expansion coefficients and the permanents themselves are treated as the variational parameters and have to be determined self-consistently. Illustrative examples of bosonic systems made of N=1000 trapped in one- and two-dimensional symmetric and asymmetric double-well traps are provided. We demonstrate that self-consistency has a great impact on the predicted physical properties of the ground and excited states and quantum phase transitions. Moreover, we show that the lack of self-consistency may lead to the physically wrong predictions. |
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