Comparison of the non-crossing perturbative approach and the generalized projection method for strongly coupled spin-fermion systems at low doping

We analyse the two-dimensional spin-fermion model in the strong-coupling re gime relevant to underdoped cuprates. We recall the set df general sum rule s that relate the moments of spectral density and the imaginary part of the fermion self-energy to the static correlation functions. We show that the two-pole approximation of the projection method satisfies the sum rules for the first four moments of the spectral density and gives an exact upper bo und for the quasiparticle energy near the band bottom. We prove that the no n-crossing approximation that is often made in perturbative considerations of the model violates the sum rule for the third moment of the spectral den sity. This leads to incorrect positioning of the lowest quasiparticle band. On the other hand, the projection method is inadequate in the weak-couplin g limit because of the approximate treatment of the kinetic energy term. We propose a generalization of the projection method that resolves this probl em, and give a fermion self-energy that behaves correctly in both the weak- and strong-coupling limits.