Effective action approach and Carlson-Goldman mode in d-wave superconductors

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We theoretically investigate the Carlson-Goldman (CG) mode in two-dimensional clean d-wave superconductors using the effective ``phase only'' action formalism. In conventional s-wave superconductors, it is known that the CG mode is observed as a peak in the structure factor of the pair susceptibility S(Ω, K) only just below the transition temperature Tc and only in dirty systems. On the other hand, our analytical results support the statement by Y.Ohashi and S.Takada, Phys.Rev.B 62, 5971 (2000) that in d-wave superconductors the CG mode can exist in clean systems down to the much lower temperatures, T ≈ 0.1 Tc. We also consider the manifestations of the CG mode in the density-density and current-current correlators and discuss the gauge independence of the obtained results.