Renormalization of the electron-spin-fluctuation interaction in the $t\text{−}{t}^{\prime }\text{−}U$ Hubbard model

We study the renormalization of the electron-spin-fluctuation (el-sp) vertex in a two-dimensional Hubbard model with nearest-neighbor $\left(t\right)$ and next-nearest-neighbor $\left(t^{\prime }\right)$ 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 $t^{\prime }=0$ 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 $t^{\prime }∕t<0$ 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 $t^{\prime }$.

Figure 1

Diagrammatic representation of $G_A(p,q)$ within linear response to $S_{\mathbf{q}}$. The thick solid lines represent dressed single-particle Green’s functions of the Hubbard model. The wavy line denotes the external perturbation in Eq. (5). The dashed line represent the Hubbard interaction $U$ and the black ellipse stands for the longitudinal spin susceptibility ${\chi }_{zz}\left(q\right)$.

Figure 2

(Color online) (a) Spin susceptibility ${\chi }_{zz}\left(q\right)$, (b) renormalized el-sp vertex $\mid \Gamma (p,q)\mid$, and (c) total pairing interaction $V_p(p,q)$ as a function of spin-fluctuation momentum transfer $\mathbf{q}$. Here $U=4$, $t^{\prime }=0$, $\mathbf{p}=(-\pi ,0)$, and the doping density $\delta =0.12$. The value of the inverse temperature $\beta$ is indicated by the shape of the symbol.

Figure 3

(Color online) The absolute values of the (complex) renormalized el-sp vertices $\mid \Gamma (p,q)\mid$ and $\mid \gamma (p,q)\mid$ vs $T$ at $U=4$ and 8 at the doping density $\delta =0.12$. We set $\mathbf{p}=(-\pi ,0)$, $\mathbf{q}=(\pi ,\pi )$, and $t^{\prime }=0$.

Figure 4

(Color online) The pairing interaction $V_p(p,q)$ vs $T$ for (a) $U=4$ and (b) 8 at the doping density $\delta =0.12$ (full symbols; the solid line is a guide to the eye). The open symbols (dashed line) show the RPA results. In (c), we show the $T$ dependence of the wave-function renormalization $Z\left(p\right)$ and in (d) the effective pairing interactions $v_p(p,q)$ between quasiparticles. In all panels, $\mathbf{p}=(-\pi ,0)$, $\mathbf{q}=(\pi ,\pi )$, and $t^{\prime }=0$.

Figure 5

(Color online) $V_p(p,q)∕Z\left(p\right)$ vs $T$ for $U=4$ and 8 at the doping density $\delta =0.12$ (full symbols; the solid line is a guide to the eye). $\mathbf{p}$ and $\mathbf{q}$ are the same as in Fig. 4.

Figure 6

(Color online) (a) Spin susceptibility ${\chi }_{zz}\left(q\right)$, (b) “damping rate” ${\gamma }_d$, and (c) el-sp vertex $\mid \Gamma (p,q)\mid$ as a function of doping density $\delta$ for $U=4$. The value of $t^{\prime }$ is indicated by the shape of the symbol. Here $\mathbf{p}=(-\pi ,0)$, $\mathbf{q}=(\pi ,\pi )$, and $\beta =4$.

Figure 7

(Color online) (a) el-sp vertex $\mid \gamma (p,q)\mid$ and (b) $V_p(p,q)∕Z\left(p\right)$ as a function of doping density $\delta$ for $U=4$. $\mathbf{p}$ and $\mathbf{q}$ are the same as in Fig. 6.