Pairing theory for strongly correlated $d$-wave superconductors

Motivated by recent proposals of correlation induced insensitivity of $d$-wave superconductors to impurities, we develop a simple pairing theory for these systems for up to a moderate strength of disorder. Our description implements the key ideas of Anderson, originally proposed for disordered $s$-wave superconductors, but in addition takes care of the inherent strong electronic repulsion in these compounds, as well as disorder induced inhomogeneities. We first obtain the self-consistent one-particle states, which capture the effects of disorder exactly, and strong correlations using Gutzwiller approximation. These “normal states,” representing the interplay of strong correlations and disorder, when coupled through pairing attractions following the path of Bardeen-Cooper-Schrieffer (BCS), produce results nearly identical to those from a more sophisticated inhomogeneous Hartree-Fock-Bogoliubov analysis that takes care of strong electronic repulsions.

Figure 1

(a)–(d) Spatial density map of $V_{\mathrm{eff}}$ and $t_{\mathrm{eff}}$ from NS and GIMT shows similar spatial anticorrelation between $V_{\mathrm{eff}}$ and $t_{\mathrm{eff}}$. They also highlight the spatial correlation in $V_{\mathrm{eff}}$. Comparison of distributions $P\left(V_{\mathrm{eff}}\right)$ in (e) and $P\left(t_{\mathrm{eff}}\right)$ in (f) for $\mathrm{NS}$ and for the full GIMT outputs at $V=1.75$. The distributions match rather well in the two calculations validating the basis of PNS calculations. We subtracted the homogeneous components of $t_{\mathrm{eff}}$ ($t_{\mathrm{eff}}(V=0)=0.459$) and $V_{\mathrm{eff}}$ ($V_{\mathrm{eff}}(V=0)=1.6$), arising from the Fock and Hartree shifts, respectively. The resulting distributions in (e) and (f) feature zero mean—this is broadly true for all $V$.

Figure 2

A schematic evolution of the inhomogeneity in space that leads to the renormalization of $V_{\mathrm{eff}}\left(i\right)$ and spatial anticorrelation between $V_{\mathrm{eff}}\left(i\right)$ and $t_{\mathrm{eff}}\left(i\right)$, upon including electronic repulsions through the Gutzwiller approximation (GA). Consider in (a) the site $i$ having a high hill of disorder potential (also assumed that $V_{i\pm \delta }=0$), that would normally yield a low ${\rho }_i$ compared to ${\rho }_{i\pm \delta }\approx {\rho }_0$, as shown in (b), rarely populating the site $i$. However, GA insures that $t_{\mathrm{eff}}$ on bonds connecting $i$ to its neighbors is enhanced, according to Eq. (5), increasing charge flow to this site. This in turn reduces the dip in ${\rho }_i$ as seen in (c), so that the corresponding $V_{\mathrm{eff}}\left(i\right)$, which would have normally produced the ${\rho }_i$ in (c), is far weaker than its bare value, shown in (a). Exactly similar arguments would yield a similar weakening of deep potential well by strong correlations.

Figure 3

Scatter plots of $V_{\mathrm{eff}}\left(i\right)$ against the bare potential $V\left(i\right)$ for (a) $V=1.0$ and (b) 2.5. The red lines are the best fit to the data. The slope of the solid line in both panels is close to the average doping ($\delta =0.2$). For $V=2.5$, the data tend to deviate from the fit for larger $\left|V\right(i\left)\right|$, signaling higher order effects.

Figure 4

Main panels: The distribution $P\left({\mathcal{R}}_{{\mathrm{\Delta }}_d}\right)$ in (a) and of $P\left({\mathcal{R}}_{\rho }\right)$ in (b) are shown for various disorder strengths. Sharply peaked nature of these distributions (with small variance) validates the PNS formalism. The left inset of panel (a) shows the PNS results for the spatial profile of ${\mathrm{\Delta }}_d$ at $V=2.5$, where as the right inset presents that of ${\mathcal{R}}_{{\mathrm{\Delta }}_d}$. The strong resemblance of the two ascertains that the small differences between GIMT and PNS outputs are not just random in space, but are highly correlated with the order parameter profile. The insets of panel (b) draw similar inferences as those in (a), but for local densities.

Figure 5

Evolution of ${\mathrm{\Delta }}_{\mathrm{OP}}$ is presented against $V$. The $V$ dependencies of both the PNS and GIMT results show nearly identical behavior. The inset shows an expanded region of the main panel establishing that the PNS findings match excellently with those from GIMT within the error bars. The results for ${\mathrm{\Delta }}_{\mathrm{OP}}$ from IMT calculations, shown by the magenta curve (forcing all Gutzwiller factors to unity, and thereby neglecting strong electronic repulsions), deviate significantly from the PNS or GIMT results. However, it still complements the plain BdG results (red dashed line) exceedingly well.

Figure 6

DOS of dSC state calculated within PNS from (a) ${\mathrm{NS}}_{\mathrm{GIMT}}$ and (b) ${\mathrm{NS}}_{\mathrm{IMT}}$. The features in (a) and (b) match very well with the findings of GIMT and IMT, respectively. DOS for the normal states (c) ${\mathrm{NS}}_{\mathrm{GIMT}}$ and (d) ${\mathrm{NS}}_{\mathrm{IMT}}$. Insets of (c) and (d) show an expanded view of the DOS at low energies. For $V⩾0.5$, ${\mathrm{NS}}_{\mathrm{IMT}}$ becomes flat near the chemical potential analogous to an Anderson insulator. Strongly correlated ${\mathrm{NS}}_{\mathrm{GIMT}}$ preserves the Van Hove singularity owing to the complex interplay of $V_{\mathrm{eff}}$ and $t_{\mathrm{eff}}$. A soft gap develops at the chemical potential for $V⩾2.0$ in both ${\mathrm{NS}}_{\mathrm{GIMT}}$ and ${\mathrm{NS}}_{\mathrm{IMT}}$.

Figure 7

Intensity plot of $|{\mathrm{\Delta }}_{\alpha \beta }|$ in the normal state eigenbasis $\alpha \text{−}\beta$ for (a) $V=1.0$ and (b) 2.5. We show $|{\mathrm{\Delta }}_{\alpha \beta }|$ in a limited range of $\alpha ,\beta$ (only the central part) for a better resolution. The presented values of $|{\mathrm{\Delta }}_{\alpha \beta }|$ are scaled by their maximum values for clarity (0.35 for $V=1.0$ and 0.3 for $V=2.5$). The near-diagonal nature of the pairing is evident for both $V$. The color scales are identical to that in Fig. 1. (c) Density plot of $|{\mathrm{\Delta }}_{\alpha \beta }|$ against ${\xi }_{\alpha }$ and ${\xi }_{\beta }$ across the full (renormalized) bandwidth for $V=2.5$. While the diagonal character of $|{\mathrm{\Delta }}_{\alpha \beta }|$ is evident, only negligible contribution to $|{\mathrm{\Delta }}_{\alpha \alpha }|$ comes from the states near band edges. (d) Accuracy of PNS (with respect to GIMT) is shown along $y$ axis, against the percentage of states paired (along $x$ axis). This accuracy, already impressive with about 10% ${\mathrm{NS}}_{\mathrm{GIMT}}$ participating in pairing, becomes better as more states included in Cooper pairing.

Figure 8

${\mathrm{\Delta }}_{ij}$ on each bond for a section of the lattice for (a) $V=2.5$ and (b) 4.5 for a specific realization of disorder. The $\pi /2$ phase difference between ${\mathrm{\Delta }}_{i,i+\widehat{x}}$ and ${\mathrm{\Delta }}_{i,i+\widehat{y}}$ survives over the entire lattice as seen in (a). The larger disorder strength of (b) still supports the $d$-wave anisotropy in most parts (highlighted by the square boundary). It also features regions of strong potential fluctuations (marked by circular boundary), where ${\mathrm{\Delta }}_{i,i+\widehat{x}}$ and ${\mathrm{\Delta }}_{i,i+\widehat{y}}$ are closer in magnitude, but only when both are vanishingly small!