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Physical review. B, Condensed matter

Exact identities in the Hubbard model andt-Jmodel

Yue Yu, Zhongyuan Zhu

Published Jun 1, 1992• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.45.12502

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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.
Abstract The extended Hubbard model can host s-wave, d-wave and p-wave superconducting phases depending on the values of the on-site and nearest-neighbour interactions. Upon detailed examination of the free energy functional of the gap in this model, we show that these symmetries are often dependent on temperature. The critical points of this functional are constrained by symmetry and allow us to formulate stringent conditions on the temperature profile of the gap function, applicable to other models as well. We discuss the finite temperature phase diagram of the extended Hubbard model, and point out the existence of symmetry transitions below T c . Understanding the nature of these transitions is crucial to assessing the symmetry of unconventional superconductors.
The existence of more than one solution for s-wave pairing in the extended Hubbard model is not often realized. This possibility was noted by Friedberg et al. [Phys. Rev B50, 10190 (1994)] in the case of two electrons on a lattice, without further analysis. In the present paper, the second solution is found also in the case of superconductivity of extended s-wave symmetry in the extended Hubbard model. The properties of both s-wave solutions are examined by mean-field methods and thresholds for their appearance are given. A possibility of the first-order transition is discovered and modifications of the phase diagram are calculated. Results of this paper in the limit of low electron density are also applicable to the bound, two-electron pairs.
Using symmetry considerations and exact diagonalization techniques, we examine the quantum numbers of the states that define the antiferromagnetic and superconducting gaps in the multiple-particle spectrum of the repulsive and attractive two-dimensional Hubbard models. Studying the behavior of the lowest-energy levels as a function of the chemical potential, we analyze all possible dopings in 10- and 16-site clusters for many values of U/t. The possibility of superconductivity in the repulsive Hubbard and in the t-J models is discussed.
We show that the exact ground state for a class of extended Hubbard models including bond-charge, exchange, and pair-hopping terms, is the Yang ``\ensuremath{\eta}-paired'' state for any nonvanishing positive value of the pair-hopping amplitude, at least when the on-site Coulomb interaction is attractive enough and the remaining physical parameters satisfy a single constraint. The ground state is thus rigorously superconducting. Our result holds on a bipartite lattice in any dimension, at any band filling, and for arbitrary electron hopping. \textcopyright{} 1996 The American Physical Society.