Resilience of $d$-wave superconductivity to nearest-neighbor repulsion

Many theoretical approaches find $d$-wave superconductivity in the prototypical one-band Hubbard model for high-temperature superconductors. At strong coupling ($U\ge W$, where $U$ is the on-site repulsion and $W=8t$ the bandwidth) pairing is controlled by the exchange energy $J=4t^2/U$. One may then surmise, ignoring retardation effects, that near-neighbor Coulomb repulsion $V$ will destroy superconductivity when it becomes larger than $J$, a condition that is easily satisfied in cuprates, for example. Using cellular dynamical mean-field theory with an exact diagonalization solver for the extended Hubbard model, we show that pairing at strong coupling is preserved, even when $V\gg J$, as long as $V\lesssim U/2$. While at weak coupling $V$ always reduces the spin fluctuations and hence $d$-wave pairing, at strong coupling, in the underdoped regime, the increase of $J=4t^2/(U-V)$ caused by $V$ increases binding at low frequency while the pair-breaking effect of $V$ is pushed to high frequency. These two effects compensate in the underdoped regime, in the presence of a pseudogap. While the pseudogap competes with superconductivity, the proximity to the Mott transition that leads to the pseudogap, and retardation effects, protect $d$-wave superconductivity from $V$.

Figure 1

Cluster and bath parametrization used in this work. Hopping terms are shown by solid lines, bath hybridization by dashed lines, and bath pairing terms by dotted lines.

Figure 2

Blue dots: Inverse frequency of the dominant peak in the antiferromagnetic susceptibility of the extended Hubbard model at half filling, as a function of the nearest-neighbor interaction $V$, for $U=16$ and $U=32$ ($t=1$). These inverse frequencies are expected to be $J^{-1}=(U-V)/4t^2$ in the large-$U$ limit.

Figure 3

$d$-wave order parameter $\psi$ obtained from the off-diagonal component of the lattice Green's function as a function of cluster doping for $U=4$, 8, and 16 and various values of $V$.

Figure 4

(a) Imaginary part of the local spin susceptibility ${\chi }^{\prime \prime }\left(\omega \right)$ for $U=8$, $V=1$ for an underdoped ($x=0.05$) and an overdoped ($x=0.20$) case. (b) In the underdoped regime ($x=0.05$) apart from a shift of the low-frequency peak to higher frequencies, it is mostly the amplitude, not the structure of the imaginary part of the plaquette antiferromagnetic spin susceptibility ${\chi }^{\prime \prime }(\mathbf{Q},\omega )$, that is slightly affected by $V$.

Figure 5

Integral of the anomalous Green's function (or Gork'ov function) $I_F\left(\omega \right)$ obtained after extrapolation to $\eta =0$ of $\omega +i\eta$ for several values of $V$ at (a) $U=8$, $x=0.05$, (b) $U=8$, $x=0.2$, and (c) $U=16$, $x=0.05$. The asymptotic value of the integral, $I_F\left(\infty \right)$, equal to the order parameter, is shown as horizontal lines. We call $I_F\left(\omega \right)$ the cumulative order parameter. The characteristic frequency ${\omega }_F$ is defined as the frequency at which $I_F\left(\omega \right)$ is equal to half of its asymptotic value. The horizontal arrow in (a) indicates how ${\omega }_F$ is obtained.

Figure 6

(a) Position of the lowest peak in the AF susceptibility ${\chi }^{\prime \prime }\left(\omega \right)$ and the strength $\chi$ of that peak in (b), for $U=8$. (c) shows the corresponding superconducting (SC) characteristic frequency ${\omega }_F$.

Figure 7

Local density of states for four values $V=1,2,3,4$ and $U=8$ for (a) underdoped $x=0.05$, and (b) overdoped $x=0.2$ regimes. The dotted line in (b) is for the normal state $V=0$, all other lines in (a) and (b) are for the superconducting state.