Conserving and Gapless Hartree-Fock-Bogoliubov theory for 3D dilute Bose gas at finite temperature

The energy spectrum for the three dimensional Bose gas in Bose-Einstein Condensation phase is calculated with Modified Hartree-Fock-Bogoliubov theory, which is both conserving and gapless. From Improved $\Phi -$% derivable theory, the diagrams needed to preserve Ward-Takahashi Identity are resummed in a systematic and nonperturbative way. The results show significant discrepancies with Popov theory at finite temperature. It is valid up to the critical temperature where the dispersion relation of the low energy excitation spectrum changes from linear to quadratic. Because of the repulsive interaction, the critical temperature has a positive shift from that of idea gas, which is in accordance with the result from the previous calculations in the uncondensed phase.