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Constraints on pairings in the Hubbard model - Comment

XZ Yan

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

We show that Shi's conclusion about the singlet pairing in the Hubbard model at strong-coupling limit [Phys. Rev. B 51, 16 619 (1995)] is wrong. The singlet pairing is related to many-particle correlated pairings. There generally is no exact identity among the singlet pairing functions. We give an intuitive argument that the nearest-neighbor d(x2-y2)-wave pairing is the predominant one in the doped Hubbard model.

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Comment on ‘‘Constraints on pairings in the Hubbard model’’

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We show that Shi's conclusion about the singlet pairing in the Hubbard model at strong-coupling limit [Phys. Rev. B 51, 16 619 (1995)] is wrong. The singlet pairing is related to many-particle correlated pairings. There generally is no exact identity among the singlet pairing functions. We give an intuitive argument that the nearest-neighbor ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is the predominant one in the doped Hubbard model. \textcopyright{} 1996 The American Physical Society.

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Philosophical Magazine B•Nov 1, 1997•1 citation
We show that Shi's conclusion about the singlet pairing in the Hubbard model at strong-coupling limit [Phys. Rev. B 51, 16 619 (1995)] is wrong. The singlet pairing is related to many-particle correlated pairings. There generally is no exact identity among the singlet pairing functions. We give an intuitive argument that the nearest-neighbor d(x2-y2)-wave pairing is the predominant one in the doped Hubbard model.
We show that Shi's conclusion about the singlet pairing in the Hubbard model at strong-coupling limit [Phys. Rev. B 51, 16 619 (1995)] is wrong. The singlet pairing is related to many-particle correlated pairings. There generally is no exact identity among the singlet pairing functions. We give an intuitive argument that the nearest-neighbor ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is the predominant one in the doped Hubbard model. \textcopyright{} 1996 The American Physical Society.
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 calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and ddd-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.
Superconducting phases of the extended Hubbard model has been investigated in the weak-coupling limit at different temperatures and doping levels in the presence of on-site, nearest-neighbour and next-nearest-neighbour couplings. The main result is the occurrence of stable pairing states of different symmetries depending on the tunable parameters of the model; attractive nearest-neighbour couplings give rise to a stable superconducting state having dxr y2 symmetry, while attractive next-nearest-neighbour interactions favour pairing states of dxy symmetry.