Spin susceptibility representation of the pairing interaction for the two-dimensional Hubbard model

Using numerical dynamic cluster quantum Monte Carlo results, we study a simple approximation for the pairing interaction of a two-dimensional Hubbard model with an on-site Coulomb interaction $U$ equal to the bandwidth. We find that with an effective temperature-dependent coupling $\overline{U}\left(T\right)$ and the numerically calculated spin susceptibility $\chi (K-K^{\prime })$, the $d$-wave pairing interaction is well approximated by $\frac{3}{2}{\overline{U}}^2\chi (K-K^{\prime })$.

Figure 1

(Color online) The coupling strength $\overline{U}$ for $U=8$ in units in which $t=1$ and a site filling $⟨n⟩=0.85$ calculated for two cluster sizes $N_c=4$ and 24.

Figure 2

(Color online) The $d$-wave eigenvalue versus $T$ obtained from the RPA form [Eq. (1)] (open symbols) and from the “exact” DCA interaction (solid symbols) for $N_c=4$ and 24.

Figure 3

(Color online) The momentum dependence of the $d$-wave eigenvector $\varphi (\mathbf{K},{\omega }_n)$ for ${\omega }_n=\pi T$ and $T=0.33$ of the approximate interaction [Eq. (1)] compared to the exact DCA interaction ${\Gamma }_{\text{even}}^{\mathit{pp}}$. $\varphi (\mathbf{K},{\omega }_n)$ has been normalized to its value at $\mathbf{K}=(\pi ,0)$.

Figure 4

(Color online) The frequency dependence of the $d$-wave eigenvector $\varphi (\mathbf{K},{\omega }_n)$ for $\mathbf{K}=(\pi ,0)$ and $T=0.33$ of the approximate interaction [Eq. (1)] compared to the exact DCA interaction ${\Gamma }_{\text{even}}^{\mathit{pp}}$. $\varphi (\mathbf{K},{\omega }_n)$ has been normalized to its value at ${\omega }_n=\pi T$.

Figure 5

Decomposition of the irreducible particle-particle vertex ${\Gamma }^{\mathit{pp}}$ into a fully irreducible vertex ${\Lambda }_{\mathrm{irr}}$ plus reducible particle-hole contributions in the cross channel. Here, $\Gamma$ denotes the full two-particle vertex and ${\Gamma }^{\mathit{ph}}$ the irreducible particle-hole vertex.

Figure 6

(Color online) Decomposition of the $d$-wave pairing interaction $V_d$ into $d$-wave projections of the fully irreducible vertex ${\Lambda }_{\mathrm{irr}}$, and a magnetic $(S=1)$ channel $3∕2{\varphi }_m$ and a charge $(S=0)$ channel $1∕2{\varphi }_d$.

Figure 7

(Color online) Coupling ${\overline{\Gamma }}^{\mathit{ph}}$ to the $S=1$ susceptibility calculated according to Eq. (11).