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arXiv:cond-mat/9810253

Superconductivity in the Attractive Hubbard Model: The Double Hubbard--I Approximation

José Jesús Rodríguez-Núñez, Haranath Ghosh

Published Oct 20, 1998• 0 citations
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Abstract

Using the Dyson equation of motion for both the diagonal one-particle Green function, $G(\vec{k},ω)$ and off--diagonal Green function, $F(\vec{k},ω)$, 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 $T$-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 $G(\vec{k},ω)$ and $F(\vec{k},ω)$ have to be calculated self-consistently. $G(\vec{k},ω)$ satisfies the first two moments while $F(\vec{k},ω)$ the first sum rule. Our {\it order parameter} $α(T)$ is given by $1/N_s \sum_{\vec{k}} ε({\vec{k}})Δ({\vec{k}})$. Due to the fact that we have a purely local attractive interaction $Δ(\vec{k})$ can be of {\it any} s--type wave. However, for a {\it pure s--wave}, for which $α(T) = 0$, we go back to the mean--field $BCS$ results, with a renormalized chemical potential. In this case, the off--diagonal Green function, $F(\vec{k},ω)$, satisfies the first two off--diagonal sum rules. We explicitly state the range of validity of our approximation.

DOI
10.48550/arxiv.cond-mat/9810253

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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 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.
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 evaluate the one-particle and double-occupied Green functions for the Hubbard model at half-filling using the moment approach of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. Our starting point is a self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ which has a single pole, $\ensuremath{\Omega}(\stackrel{\ensuremath{\rightarrow}}{k}),$ with spectral weight, $\ensuremath{\alpha}(\stackrel{\ensuremath{\rightarrow}}{k}),$ and quasiparticle lifetime, $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{k})$ [J. J. Rodr\'{\i}guez-N\'u\~nez and S. Schafroth, J. Phys. Condens. Matter 10, L391 (1998); J. J. Rodr\'{\i}guez-N\'u\~nez, S. Schafroth, and H. Beck, Physica B (to be published); (unpublished)]. In our approach, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes the central feature of the many-body problem and due to three unknown $\stackrel{\ensuremath{\rightarrow}}{k}$ parameters we have to satisfy only the first three sum rules instead of four as in the canonical formulation of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. This self-energy choice forces our system to be a non-Fermi liquid for any value of the interaction, since it does not vanish at zero frequency. The one-particle Green function, $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ shows the fingerprint of a strongly correlated system, i.e., a double peak structure in the one-particle spectral density, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ vs $\ensuremath{\omega}$ for intermediate values of the interaction. Close to the Mott insulator-transition, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes a wide single peak, signaling the absence of quasiparticles. Similar behavior is observed for the real and imaginary parts of the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}).$ The double-occupied Green function, ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega}),$ has been obtained from $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ by means of the equation of motion. The relation between ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ and the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ is formally established and numerical results for the spectral function of ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ ${\ensuremath{\chi}}^{(2)}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})\ensuremath{\equiv}\ensuremath{-}(1/\ensuremath{\pi}){limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}{0}^{+}}\mathrm{Im}[{G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})],$ are given. Our approach represents the simplest way to include (1) lifetime effects in the moment approach of Nolting, as shown in the paper, and (2) Fermi or/and marginal Fermi liquid features as we discuss in the conclusions.
We study excitation-spectrum and normal-state properties of the two-dimensional attractive Hubbard model using the conserving, self-consistent T-matrix formalism in the intermediate coupling regime and at low electron concentration. Numerical results are presented for one-particle and two-particle excitation spectra, the one-particle momentum distribution, the chemical potential, and the static spin susceptibility. For a coupling strength of U/t=4.0, the one-particle spectral function, A(k,\ensuremath{\omega}), shows two peaks of different weights. One peak can be associated with pair formation, whereas the other corresponds to renormalized quasiparticle excitation. It turns out that the two-band feature is reasonably well described by an ansatz for A(k,\ensuremath{\omega}), which satisfies the first four frequency moments.
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.