Collective modes of spin, density, phase and amplitude in exotic superconductors

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding band, where the electrons experience besides the long-range Coulomb repulsion an on-site Hubbard U repulsion and a nearest-neighbour attractive interaction. {}From numerical calculations we see, that the collective mode spectrum now becomes particularly rich. Several branches can occur below the continuum of quasi-particle excitations, corresponding to order-parameter fluctuations of various symmetries of pairing, and collective spin-density fluctuations. {}From the collective mode softening near the nesting vectors it is concluded, that in the d-wave paired state an instability occurs toward the formation of a spin-density wave.