Spontaneous U(1) symmetry breaking and Bose--Einstein condensation

Based on Bogoliubov's truncated Hamiltonian H_B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term \sqrtV(\lambda a_0+\lambda^* a_0^+) to H_B, we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose--Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schr\"odinger eigenvalue equation and using the self-consistent condition.