Goldstone Mode in Charged Superconductivity: Theoretical Studies of the Carlson-Goldman Mode and Effects of the Landau Damping in the Superconducting State

We investigate the collective phase oscillations in superconductivity, i.e., the Carlson-Goldman mode in the charged system and the phason in the neutral, focusing on their velocities and how they appear in the structure function of the phase fluctuation of the order parameter. Our microscopic theory can cover from the clean to the dirty system, and furthermore it can treat the Landau damping correctly. Based on our formulation, we obtain following results: (1) The velocity of the phason is proportional to (T_C - T)^1/6 (T ≲ T_C) in the clean system, though it has been believed as (T_C - T)^1/4 (T_C: superconducting transition temperature). (2) The structure function has a central peak which grows as T → T_C. This phenomenon is related to the central peak problem in the field of the structural phase transition. (3) The Carlson-Goldman mode can be observed by a tunneling experiment in case of the dirty system, while not in the clean. It is in contrast to the previous prediction. All of these results are found to be deeply related to the presence of the Landau damping as well as the screening effect, both of which are caused by the quasi-particles excited thermally.