Vortex Corrections in Antiferromagnetic Spin Fluctuation Theories.

The issue of the vertex corrections in the spin-fermion model for cuprates has attracted a substantial interest over the last few years. The fully renormalized spin-fermion vertex, g full appears as the overall factor in the effective pairing interaction mediated by spin fluctuations: G(q ,v) 5g full 2 x(q ,v) where x(q ,v) is the dynamical spin susceptibility, and q and v are transferred momentum and frequency. Near the antiferromagnetic instability, the susceptibility is peaked at q5Q5(p ,p) and v50 which gives rise to an attraction in the dx22y2 pairing channel. However, there exists the possibility that in the same range of dopings where the susceptibility is peaked, vertex corrections substantially reduce the spin-fermion vertex. This possibility stems from the Ward identity in the magnetically ordered state, which implies that the fully renormalized vertex in fact vanishes at q5Q and depends linearly on the deviation of the momentum from Q . In this situation, g full 2 fully compensates the enhancement of the susceptibility near Q , which in turn eliminates the attraction in the d-wave channel. In the magnetically disordered state, the vertex indeed remains finite at q5Q , but it may still be substantially reduced compared to its bare value over some range of doping levels. Several groups have studied whether the precursors of a strong vertex reduction can be seen already at optimal doping. At this doping level, the Fermi surface is large which implies that the precursors of the spin-density-wave state, which appear at strong coupling, have not been formed yet. In this situation, one can expect the expansion in the spinfermion coupling constant, g , to be at least qualitatively valid. One of us and Altshuler et al. have computed the vertex corrections in the spin-fermion model restricting with the corrections which logarithmically depend on the correlation length and assuming that the correlation length is large. They found that the total vertex correction at small/ moderate doping is quite small numerically ~see below! and also has a positive sign, i.e., vertex corrections actually slightly increase the pairing interaction. A small positive vertex correction was also found numerically by another one of us.