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Physical review. B, Condensed matter

Constraints ons-wave pairing in the Hubbard model

Shoucheng Zhang

Published Jul 1, 1990• 29 citations
Publisher

Abstract

We present an exact identity between the on-site and the extended s-wave pairing amplitudes in the Hubbard model. We prove that the extended s-wave pairing amplitude vanishes identically at half filling for both signs of U. Away from half filling, the existence of an on-site s-wave pairing is a necessary and sufficient condition for the existence of an extended s-wave pairing. This result gives rise to constraints on the possible symmetries of the superconducting gap in the Hubbard model.

DOI
10.1103/physrevb.42.1012

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We present an exact identity between the on-site and the extended s-wave pairing amplitudes in the Hubbard model. We prove that the extended s-wave pairing amplitude vanishes identically at half filling for both signs of U. Away from half filling, the existence of an on-site s-wave pairing is a necessary and sufficient condition for the existence of an extended s-wave pairing. This result gives rise to constraints on the possible symmetries of the superconducting gap in the Hubbard model.
Exact identities among the nth extended s-wave-pairing amplitudes in the Hubbard model are presented. They provide more constraints on the possible symmetries of the superconducting gap in the Hubbard model than that between the on-site and the first extended s-wave pairings pointed out by Zhang. For the t-J model, we also find an exact identity between the on-site and the first extended s-wave-pairing amplitudes, which is essentially consistent with that in the Hubbard model. Besides the s-wave pairings, we discuss the constraints on the other pairing amplitudes.
Pairing operators of electrons with an arbitrary distance on a square lattice are defined in each possible wave channel, s, p, dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2, dxy{\mathit{d}}_{\mathit{x}\mathit{y}}dxy​, and gx3{\mathit{g}}_{\mathit{x}}^{3}gx3​y-xy3{\mathit{xy}}^{3}xy3. Exact identities among pairing amplitudes in the Hubbard model are presented, giving powerful constraints on the nature of pairings. At half filling, some of the pairing amplitudes, including those of the second-shortest pairing distance, vanish identically in each wave channel except the p wave. There are correlations among pairing amplitudes away from half filling, as well as nonidentically vanishing ones at half filling. Even if the pairing amplitudes vanish up to a certain distance, those of larger distances can still possibly be nonzero. The order of magnitude is estimated for the pairine amplitudes, indicating that those of large distances can by no means be neglected. The scheme is also used in the extended Hubbard model including nearest-neighbor density-density interaction and that including next-nearest-neighbor hopping. In that including nearest-neighbor interaction, the features of pairing amplitudes are just the same as in the original Hubbard model at half filling, and are also similar away from half filling, but with the difference that the amplitude is now dependent on the nearest-neighbor interaction. In the model including next-nearest-neighbor hopping, the features are quite different, the constraint identities are more underconstrained, and the amplitude has no monotonic dependence upon the pairing distance if the next-nearest-neighbor hopping has an opposite sign to the nearest-neighbor hopping as required physically. The same scheme is also used to discuss the possible pairings in the coupled two-plane Hubbard model and in the infinite-layer Hubbard model.
We investigate the possibility of superconductivity in the large-UUU limit of the Hubbard model near half filling, using a slave-boson representation. The local constraint is treated in a novel way, incorporating short-range fluctuations in addition to the global constraint. We find slave-boson pairing and a superconducting extended sss-wave phase at hole concentrations $\ensuremath{\delta}$-above 4% and below 32%.
The authors study a simple model for superconductivity based on an extended Hubbard model with on-site repulsive and inter-site attractive interaction for arbitrary electron density. The two-dimensional square lattice d-wave pairing is most stable for a nearly half-filled band. p-wave and then extended s-wave pairing become stable upon decreasing band filling. Competition between the various anisotropic pairings and antiferromagnetism as well as the effect of next-nearest-neighbour hopping is discussed. The authors relate the present model to high-T c superconducting oxides.
The phase diagram of the unconstrained t−Jt-Jt−J model is calculated using the random phase approximation. It is found that the extended sss and the dx2−y2d_{x^2-y^2}dx2−y2​-channels are {\em not} degenerate near half filling. Extended sss-pairing with a low TcT_cTc​ occurs only for a band containing less then 0.4 electrons or holes per unit cell, whereas in a large region around half-filling ddd-wave pairing is the only stable superconducting solution. At half filling superconductivity is suppressed due to the formation of the anti-ferromagnetic Mott-Hubbard insulating state. By extending the analysis to the unconstrained t−t′−Jt-t'-Jt−t′−J model, it is proven that, if a Fermi surface is assumed similar to the one that is known to exist in cuprous oxide superconductors, the highest superconducting TcT_cTc​ is reached for about 0.70.70.7 electron per site, whereas the anti-ferromagnetic solution still occurs for 111 electron per site. It is shown, that the maximum ddd-wave superconducting mean field transition temperature is half the maximum value that the Neèl temperature can have in the Mott-insulating state.