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arXiv:cond-mat/9806264 · Physical Review Letters

d -Wave Pairing in the Presence of Long-Range Coulomb Interactions

Gökhan Esirgen, H.-B. Schüttler, N. E. Bickers

Published Feb 8, 1999• 26 citations
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Abstract

The one-band extended Hubbard model in two dimensions near band-filling $\frac{1}{2}$ is solved in the fluctuation exchange approximation, including the long-range ( $1/r$) part of the Coulomb interaction, up to 4th-neighbor distance. Our results suggest that ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing in the Hubbard model is robust against the inclusion of long-range Coulomb interactions with moderate 1st-neighbor repulsion strength ${V}_{1}$. ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing is suppressed only at large ${V}_{1}$ ( $\ensuremath{\gtrsim}0.25U--0.4U$), due to incipient charge density wave instabilities.

DOI
10.1103/physrevlett.82.1217

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The one-band extended Hubbard model in two dimensions near band-filling 12\frac{1}{2}21​ is solved in the fluctuation exchange approximation, including the long-range ( 1/r1/r1/r) part of the Coulomb interaction, up to 4th-neighbor distance. Our results suggest that dx2\ensuremath−y2{d}_{{x}^{2}{\ensuremath{-}y}^{2}}dx2\ensuremath−y2​ pairing in the Hubbard model is robust against the inclusion of long-range Coulomb interactions with moderate 1st-neighbor repulsion strength V1{V}_{1}V1​. dx2\ensuremath−y2{d}_{{x}^{2}{\ensuremath{-}y}^{2}}dx2\ensuremath−y2​ pairing is suppressed only at large V1{V}_{1}V1​ ( \ensuremath≳0.25U−−0.4U\ensuremath{\gtrsim}0.25U--0.4U\ensuremath≳0.25U−−0.4U), due to incipient charge density wave instabilities.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g{B}_{2g}B2g​ (dxy{d}_{xy}dxy​) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN{V}_{\mathrm{NN}}VNN​, the latter induces pairing in other symmetry channels, including, e.g., A2g{A}_{2g}A2g​ (ggg-wave), nodal A1g{A}_{1g}A1g​ (extended sss-wave), or nodal Eu{E}_{u}Eu​ (ppp-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0{V}_{\mathrm{NN}}<0VNN​<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu{E}_{u}Eu​ (ppp-wave) spin-triplet or B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
It is rigorously shown that the two-dimensional Hubbard model with narrow bands (including next-nearest-neighbor hopping, etc.) does not exhibit ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing long-range order at any nonzero temperature. This kind of pairing long-range order will also be excluded at zero temperature if an excited energy gap opens in the charge excitation spectrum of the system. These results hold true for both repulsive and attractive Coulomb interactions and for any electron fillings, and are consistent with quantum Monto Carlo calculations.
Dynamic cluster quantum Monte Carlo calculations for a doped two-dimensional extended Hubbard model are used to study the stability and dynamics of ddd-wave pairing when a nearest-neighbor Coulomb repulsion VVV is present in addition to the on-site Coulomb repulsion UUU. We find that ddd-wave pairing and the superconducting transition temperature Tc{T}_{c}Tc​ are only weakly suppressed as long as VVV does not exceed U/2U/2U/2. This stability is traced to the strongly retarded nature of pairing that allows the ddd-wave pairs to minimize the repulsive effect of VVV. When VVV approaches U/2U/2U/2, large momentum charge fluctuations are found to become important and to give rise to a more rapid suppression of ddd-wave pairing and Tc{T}_{c}Tc​ than for smaller VVV.
We explore how the superconductivity arising from the on-site electron-electron repulsion changes when the repulsion is made long-ranged, 1∕r1∕r1∕r-like interaction by introducing an extended Hubbard model with the repulsion extended to distant (12th) neighbors. With a simplified fluctuation-exchange approximation, we have found for the square lattice that: (i) As the band filling becomes dilute enough, the charge susceptibility becomes comparable with the spin susceptibility, where ppp- and then sss-wave pairings become relatively dominant, in agreement with the result for the electron gas by Takada, while (ii) the ddd wave, which reflects the lattice structure, dominates well away from the half-filling. All of these can be understood in terms of the spin and charge structures along with the shape and size of the Fermi surface.
Many theoretical approaches find ddd-wave superconductivity in the prototypical one-band Hubbard model for high-temperature superconductors. At strong coupling ($U\ensuremath{\ge}W$, where UUU is the on-site repulsion and W=8tW=8tW=8t the bandwidth) pairing is controlled by the exchange energy J=4t2/UJ=4{t}^{2}/UJ=4t2/U. One may then surmise, ignoring retardation effects, that near-neighbor Coulomb repulsion VVV will destroy superconductivity when it becomes larger than JJJ, 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 $V\ensuremath{\gg}J$, as long as $V\ensuremath{\lesssim}U/2$. While at weak coupling VVV always reduces the spin fluctuations and hence ddd-wave pairing, at strong coupling, in the underdoped regime, the increase of $J=4{t}^{2}/(U\ensuremath{-}V)$ caused by VVV increases binding at low frequency while the pair-breaking effect of VVV 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 ddd-wave superconductivity from VVV.
In order to identify the most favorable situation for superconductivity in the repulsive single-band Hubbard model, we have studied instabilities for d-wave pairing mediated by antiferromagnetic spin fluctuations and p-wave pairing mediated by ferromagnetic fluctuations with the fluctuation exchange approximation in both two dimensions and three dimensions. By systematically varying the band filling and band structure we have shown that (i) d pairing is stronger in two dimensions than in three dimensions and (ii) p pairing is much weaker than d pairing.