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arXiv:cond-mat/9710047

The Two - Dimensional Attractive Hubbard Model: Highly Non-Linear Superconductivity With Sum Rules

José Jesús Rodríguez-Núñez, Claudette E. Cordeiro, A. Delfino

Published Oct 4, 1997• 0 citations
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Abstract

We use the moment approach of Nolting (exact sum rules) (Z. Physik 255, 25 (1972)) for the attractive Hubbard model in the superconducting phase. Our diagonal and off - diagonal spectral functions are constructed and evaluated with the sum rules. They reduce to the $BCS$ limit for weak interaction. However, the presence of correlations modify the $BCS$ picture dramatically. For example, due to the presence of correlations we have postulated a three - pole ansatz for the diagonal Green function, $G(\vec{k},ω)$, while the off - diagonal one, $B(\vec{k},ω)$, is supposed to have two poles. In the paper we present results for the three spectral weights of the diagonal Green function, $α_j(\vec{k})$, j = 1,2,3. Our results compare reasonably well with more elaborated auto - consistent highly non - linear equations (double fluctuation calculations in the $T-$ Matrix approach of one of the authors). Then, the physical picture which emerges is that the lower Hubbard band is split due to the superconducting gap and the upper Hubbard band remains mostly unmodified.

DOI
10.48550/arxiv.cond-mat/9710047

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We use the moment approach of Nolting (exact sum rules) (Z. Physik 255, 25 (1972)) for the attractive Hubbard model in the superconducting phase. Our diagonal and off - diagonal spectral functions are constructed and evaluated with the sum rules. They reduce to the BCSBCSBCS limit for weak interaction. However, the presence of correlations modify the BCSBCSBCS picture dramatically. For example, due to the presence of correlations we have postulated a three - pole ansatz for the diagonal Green function, G(k⃗,ω)G(\vec{k},ω)G(k,ω), while the off - diagonal one, B(k⃗,ω)B(\vec{k},ω)B(k,ω), is supposed to have two poles. In the paper we present results for the three spectral weights of the diagonal Green function, αj(k⃗)α_j(\vec{k})αj​(k), j = 1,2,3. Our results compare reasonably well with more elaborated auto - consistent highly non - linear equations (double fluctuation calculations in the T−T-T− Matrix approach of one of the authors). Then, the physical picture which emerges is that the lower Hubbard band is split due to the superconducting gap and the upper Hubbard band remains mostly unmodified.
Using the Dyson equation of motion for both the diagonal one-particle Green function, G(k⃗,ω)G(\vec{k},ω)G(k,ω) and off--diagonal Green function, F(k⃗,ω)F(\vec{k},ω)F(k,ω), at the level of the Hubbard--I decoupling scheme, we have found that they have four poles symmetric in pairs, justifying a more elaborated calculation done by the Zürich group by means of the TTT-Matrix approach (Pedersen et al, Z. Physik B {\bf 103}, 21 (1997)) and the moment approach of Nolting (Z. Physik {\bf 255}, 25 (1972)). We find that the energy spectra and the weights of G(k⃗,ω)G(\vec{k},ω)G(k,ω) and F(k⃗,ω)F(\vec{k},ω)F(k,ω) have to be calculated self-consistently. G(k⃗,ω)G(\vec{k},ω)G(k,ω) satisfies the first two moments while F(k⃗,ω)F(\vec{k},ω)F(k,ω) the first sum rule. Our {\it order parameter} α(T)α(T)α(T) is given by 1/Ns∑k⃗ε(k⃗)Δ(k⃗)1/N_s \sum_{\vec{k}} ε({\vec{k}})Δ({\vec{k}})1/Ns​∑k​ε(k)Δ(k). Due to the fact that we have a purely local attractive interaction Δ(k⃗)Δ(\vec{k})Δ(k) can be of {\it any} s--type wave. However, for a {\it pure s--wave}, for which α(T)=0α(T) = 0α(T)=0, we go back to the mean--field BCSBCSBCS results, with a renormalized chemical potential. In this case, the off--diagonal Green function, F(k⃗,ω)F(\vec{k},ω)F(k,ω), satisfies the first two off--diagonal sum rules. We explicitly state the range of validity of our approximation.
We constructed the one-particle spectral functions (diagonal and off-diagonal) which reproduce BCS for weak coupling and which take into account the effect of correlations on superconductivity in the attractive Hubbard model. The diagonal spectral function is composed of three peaks and the off-diagonal one is composed of two peaks. This ansatz satisfies the sum rules for the first six moments. Our solutions are valid for intermediate coupling, i.e., for U/t ≈ −4.0. Our set of analytical equations for the unknown variables is self-consistent and has been solved numerically in lowest order of the order parameter. As a result, we obtain that the presence of the third band, or upper Hubbard band, strongly renormalizes the two lower bands, making that the energy gap be k-dependent while the order parameter is pure s-wave. This shows that the order parameter and the gap are two different quantities.
We have postulated a single pole for the self-energy looking for the consequences on the one-particle Green function, G(k,w), in the Hubbard model. We find that G(k,w) satisfies the first two sum rules or moments of Nolting (Z. Physik 225, 25 (1972)) for any values of the two unknown parameters of the self-energy. In order to find these two parameters we have used the third and four sum rules of Nolting. G(k,w) turns out to be identical to the one of Nolting which is beyond a Hubbard-I solution. Furthermore, we have also postulated a two-pole Ansatz for the self-energy which is equivalent to a three-pole ansatz for G(k,w). In the later case, we have fixed two of the frequencies. This Green function is also beyond the Hubbard-I solution. We present numerical results for the band structure and the spectral weights.
In this work, a two-dimensional one-band Hubbard model is investigated within a two-pole approximation. The model presents a non-local attractive potential U(U<0)U (U<0)U(U<0) that allows the study of d-wave superconductivity and also includes hopping up to second-nearest-neighbors. The two-pole scheme has been proposed to improve the Hubbard-I approximation. The analytical results show a more complex form for the gap Δ(T)Δ(T)Δ(T), when compared to the one obtained in the latter approximation. Indeed, new anomalous correlation functions associated with the superconductivity are involved in the calculation of Δ(T)Δ(T)Δ(T). Numerical results in a range of temperatures are presented. Moreover, the structure of the quasiparticle bands and the topology of the Fermi surface are studied in detail in the normal state. Connections with some experimental results are also included.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.