Pairing interaction in the two-dimensional Hubbard model studied with a dynamic cluster quantum Monte Carlo approximation

A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a two-dimensional Hubbard model with a near neighbor hopping $t$ and an onsite Coulomb interaction $U$. The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as $(\cos k_x–\cos k_y)$ over most of the Brillouin zone and its frequency dependence is determined by the $S=1$ particle-hole continuum which for large $U$ varies as several times $J$. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by ${\left(2J\right)}^{-1}$. The strength of the pairing interaction measured by the size of the $d$-wave eigenvalue peaks for $U$ of order of the bandwidth $8t$. It is found to increase as the system is underdoped.

Figure 1

(Color online) The single particle density of states $N\left(\omega \right)$ versus $\omega$ for $U=8t$, $N_c=16$, and various values of the temperature $T$ for site fillings of (a) $⟨n⟩=1.0$, (b) $⟨n⟩=0.90$.

Figure 2

The Bethe-Salpeter equation for the particle-particle channel showing the relationship between the four-point vertex $\Gamma$ and the particle-particle irreducible vertex ${\Gamma }^{pp}$. The solid lines are dressed single particle Green’s functions.

Figure 3

(Color online) In the DCA the Brillouin zone is divided into $N_c$ cells each represented by a cluster momentum $\mathbf{K}$. Irreducible quantities such as the single-particle self-energy $\Sigma$ and two-particle irreducible vertex $\Gamma$ are constructed from coarse-grained propagators $\overline{G}\left(\mathbf{K}\right)$ that are averaged over the momenta ${\mathbf{k}}^{\prime }$ within the cell represented by $\mathbf{K}$. The cluster momenta $\mathbf{K}$ for a four-site cluster are shown on the left and those for a 24-site cluster on the right.

Figure 4

(Color online) (a) The $d_{x^2-y^2}$ eigenvalue ${\lambda }_d\left(T\right)$ versus $T∕t$ for $U=4t$, $8t$, and $12t$ and $⟨n⟩=0.90$. (b) The $d_{x^2-y^2}$ eigenvalue ${\lambda }_d\left(T\right)$ versus $U∕t$ for $T=0.15t$ and $⟨n⟩=0.90$.

Figure 5

(Color online) The $d_{x^2-y^2}$ eigenvalue ${\lambda }_d\left(T\right)$ versus $T∕t$ for various band fillings $⟨n⟩$ for $U∕t=6$. The dashed line represents the leading eigenvalue ${\lambda }_{AF}$ in the $\mathbf{Q}=(\pi ,\pi )$, $S=1$ particle-hole channel at half filling.

Figure 6

(Color online) The $d_{x^2-y^2}$ eigenvector ${\Phi }_d(\mathbf{K},{\omega }_n)$ at ${\omega }_n=\pi T$, normalized to its value at $\mathbf{K}=(\pi ,0)$, versus $\mathbf{K}$ for $U∕t=8$, band filling $⟨n⟩=0.9$ and $T∕t=0.22$. In the main figure, the $\mathbf{K}$ points move along the dashed line shown in Fig. 3. The inset shows the behavior of ${\Phi }_d$ when $\mathbf{K}$ varies along the $k_x$ axis.

Figure 7

(Color online) The Matsubara frequency dependence of ${\Phi }_d(\mathbf{K},{\omega }_n)∕{\Phi }_d(\mathbf{K},\pi T)$ with $\mathbf{K}=(\pi ,0)$ for (a) $U∕t=4$, (b) $U∕t=8$, and (c) $U∕t=12$ calculated for $N_c=4$. Here ${\omega }_n=(2n+1)\pi T$ with $T∕t=0.125$ and the band filling $⟨n⟩=0.90$. Also shown is the frequency dependence of the normalized spin susceptibility $2\chi (\mathbf{Q},{\omega }_m)∕[\chi (\mathbf{Q},0)+\chi (\mathbf{Q},2\pi T)]$ for $\mathbf{Q}=(\pi ,\pi )$.

Figure 8

(Color online) The pairing interaction $V_d$ versus temperature for different dopings for $U=8t$ calculated for $N_c=4$.

Figure 9

(Color online) Plot of $P_{d0}\left(T\right)$ versus $T$ for $U=8t$ and $⟨n⟩=1$ showing the effect of the opening of the Mott-Hubbard gap.