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Secondly for the simplest model which permits d-wave pairing there exists a critical interaction strength for the formation of a two-body bound state on the empty lattice while this threshold is zero for s-wave pairing. This eliminates the possibility for a density induced BCS}BE crossover in this model. Thirdly the d-wave symmetry has important implications for both the excitation spectrum and the momentum distribution since at least for the chemical potential greater than the critical (or crossover) value the excitation spectrum is gapless in certain directions.

69) where 4 exp ! e. is the energy of the bound state which emerges in this case due to the electron}phonon interaction. 70) contains the quantity characterizing the energy of phonons instead of the bandwidth =. 27). The latter expression is valid for all values of o and all interaction strengths. The former applies only for weak coupling and low $ densities o ; . 66) must be used. For o ; the $ $ $ charge carriers form an `adiabaticallya slow subsystem, and the retardation of indirect interaction can be disregarded as is shown in Fig.

5, even quite sophisticated approximations may not be su$cient for 2D systems. 1. e. in the normal state. The ¹ matrix consists of the sum of all particle}particle ladder diagrams (see Fig. k)G(i , k) . M. Loktev et al. / Physics Reports 349 (2001) 1}123 43 Fig. 12. Diagrammatic representation of Eq. 1) taken from [99]. The ¹-matrix contains the repeated scattering of two-particles. Recall that , and are the fermionic (odd) and bosonic (even) Matsubara frequencies, L K respectively. This approximation is expected to be accurate in the low density limit when one can neglect particle}hole scattering and is valid when the product of the Fermi momentum, k and the $ scattering length, a is small.