Antagonistic effects of nearest-neighbor repulsion on the superconducting pairing dynamics in the doped Mott insulator regime

The nearest-neighbor superexchange-mediated mechanism for $d_{x^2-y^2}$ superconductivity in the one-band Hubbard model faces the challenge that nearest-neighbor Coulomb repulsion can be larger than superexchange. To answer this question, we use cellular dynamical mean-field theory (CDMFT) with a continuous-time quantum Monte Carlo solver to determine the superconducting phase diagram as a function of temperature and doping for on-site repulsion $U=9t$ and nearest-neighbor repulsion $V=0,2t,4t$. In the underdoped regime, $V$ increases the CDMFT superconducting transition temperature $T_c^d$ even though it decreases the superconducting order parameter at low temperature for all dopings. However, in the overdoped regime $V$ decreases $T_c^d$. We gain insight into these paradoxical results through a detailed study of the frequency dependence of the anomalous spectral function, extracted at finite temperature via the MaxEntAux method for analytic continuation. A systematic study of dynamical positive and negative contributions to pairing reveals that even though $V$ has a high-frequency depairing contribution, it also has a low frequency pairing contribution since it can reinforce superexchange through $J=4t^2/(U-V)$. Retardation is thus crucial to understanding pairing in doped Mott insulators, as suggested by previous zero-temperature studies. We also comment on the tendency to charge order for large $V$ and on the persistence of $d$-wave superconductivity over extended-$s$ or $s+d$ wave.

Figure 1

A, B, C: Superconducting phase diagram as a function of temperature and doping for different values of $V$. Color represents the value of the superconducting order parameter ${\phi }_{SC}$ for a given doping and temperature. The black points are the data points. The color maps interpolate linearly between these points. The colored lines with dots give the transition temperature. D, E, F: Color gives the difference between ${\phi }_{SC}$ at two different values of $V$. For example, the panel D gives the difference between ${\phi }_{SC}$ at $V=2$ and at $V=0$. The two colored solid lines on each plot indicate the transition temperature with the same color coding as that on the panels A, B, and C. The dashed lines marks the dopings where the superconducting ${\phi }_{SC}$ is maximal for given temperatures.

Figure 2

Hole doping evolution of the anomalous spectral function and the cumulative order parameter for $\beta =100$ and $V=0$. The positive-frequency part of the anomalous spectral weight is in the panels A and B: A for the underdoped regime and B for the overdoped regime. The vertical dashed color lines indicate the superconducting gap ${\mathrm{\Delta }}_{SC}$ extracted from the local density of states, while the black dot-dashed lines are at $J=4t^2/U$. The panels C and D display the cumulative order parameter obtained from the integral of the corresponding anomalous spectral weight on top. The horizontal dotted color lines in the panels C and D indicate the values taken by the superconducting order parameter ${\phi }_{SC}$.

Figure 3

Pairing dynamics of the attractive Hubbard model for $U=-9,V=0$ and $\beta =100$. Panel A: anomalous spectral function. Panel B: cumulative order parameter. Red dashed line: asymptotic value of the cumulative order parameter.

Figure 4

Difference in critical temperature divided by the difference in $J$ resulting from a change in $V$. Three cases, $V=0$ to 2, $V=2$ to 4, and $V=0$ to 4, are shown. While the error bars are quite large (coming from the error bars on the critical temperatures), the dark violet area corresponds to the zone where the three cases match within these error bars.

Figure 5

Analog to Fig. 1 but where the color maps refer to the strength of pair-forming processes ${\mathcal{C}}_{SC}^{+}$ [Eq. (15)] instead of the superconducting order parameter. Solid lines: critical temperatures for given dopings. Dashed lines: dopings where the superconducting order parameter is maximal for given temperatures.

Figure 6

Analog to Fig. 1 but where the color maps refer to the strength of pair-breaking processes ${\mathcal{C}}_{SC}^{-}$ [Eq. (16)] instead of the superconducting order parameter. Solid lines: critical temperatures for given dopings. Dashed lines: dopings where the superconducting order parameter is maximal for given temperatures.

Figure 7

Infinite-frequency contribution to the real part of the anomalous self-energy for different values of $V$ (the color code is the same as the one used previously). A: As a function of hole doping at $\beta =100$. B: As a function of temperature at 4% hole doping. Dashed lines: anomalous Fock contributions [Eq. (21)] for given values of $V$.

Figure 8

Analog to the panels A, B, and C of Fig. 1 but where the color maps refer to the double occupancy $D$ instead of the superconducting order parameter. Solid lines: critical temperatures for given dopings. Dashed lines: dopings where the superconducting order parameter is maximal for given temperatures. The double occupancy has been artificially put to zero outside the convex hull of the data points for the sake of clarity.