Renormalization of the electron-spin-fluctuation interaction in the t- t′ -U Hubbard model

We study the renormalization of the electron-spin-fluctuation (el-sp) vertex in a two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t′) hopping by a quantum Monte Carlo calculation. We distinguish between el-sp vertices involving interacting particles and quasiparticles, i.e., we separate the renormalization of the vertex from that of the quasiparticle residue 1 Z. We show that for t′ =0 the renormalized el-sp vertex, not dressed by 1 Z, decreases with decreasing temperature at all momentum transfers. As a consequence, the effective pairing interaction mediated by antiferromagnetic spin fluctuations is reduced due to vertex corrections. The inclusion of a finite t′ t<0 increases the Landau damping rate of spin fluctuations, especially in the overdoped region. The increased damping rate leads to smaller vertex corrections, in agreement with earlier diagrammatic calculations. Still, corrections reduce the spin-fermion vertex even at finite t′.