Effective Lagrangian approach to the trapped Bose gases at low temperatures

We adopt an effective-field-theory view to describe the low-energy excitations of trapped Bose gases, which allows a direct and systematic way to investigate the consequences of spontaneous symmetry breaking. The derivation of the effective Lagrangian can incorporate various approximations and can reproduce the results obtained by the standard hydrodynamic approach. Based on the effective Lagrangian, we calculate the energy spectrum and Matsubara Green's function of trapped one-dimensional Bose gases with delta-function repulsive interaction allowing the comparison of various results obtained by different approaches. We also analytically calculate the finite-temperature correlation function of trapped two-dimensional (2D) Bose gases. The calculation scheme can be easily extended to higher dimensions. We find that particle interactions will always decrease the coherence length of the condensate in 2D, confirming recent numerical results. The validation of various asymptotic expressions are given.