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.
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.
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 Matter 10, L391 (1998); J. J. Rodr\'{\i}guez-N\'u\~nez, S. Schafroth, and H. Beck, Physica B (to be published); (unpublished)].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.
Using the Dyson equation of motion for both the diagonal one-particle Green function, G(k,ω) and off--diagonal Green function, 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 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(k,ω) and F(k,ω) have to be calculated self-consistently. G(k,ω) satisfies the first two moments while F(k,ω) the first sum rule. Our {\it order parameter} α(T) is given by 1/Ns∑kε(k)Δ(k). Due to the fact that we have a purely local attractive interaction Δ(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(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 explore the effect of the self-energy, , having a single pole, , with spectral weight and quasi-particle lifetime , on the density of states. We obtain the set of parameters , , and by means of the moment approach (exact sum rules) of Nolting. Due to our choice of self-energy, the system is not a Fermi liquid for any value of the interaction, a result which also holds in the moment approach of Nolting without lifetime effects. Our self-energy satisfies the Kramers - Kronig relationships since it is analytic in one of the complex half-planes. By increasing the value of the local interaction, , at half-filling , there is a transition from a paramagnetic metal to a paramagnetic insulator (a Mott metal - insulator transition) for values of of the order of ( W is the bandwidth) which is in agreement with numerical results for finite lattices and for an infinite number of dimensions . These results expose the main weakness of the spherical approximation of Nolting: a finite gap for any finite value of the interaction, i.e., an insulator for any finite value of . Lifetime effects are absolutely indispensable to making our scheme work better than that based on improving the narrowing band factor, , beyond that obtained from the spherical approximation of Nolting.
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(k,ω), while the off - diagonal one, 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(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.
We derive exact expressions for the zeroth and the first three spectral moment sum rules for the retarded Green's function and for the zeroth and the first spectral moment sum rules for the retarded self-energy of the inhomogeneous Bose-Hubbard model in nonequilibrium, when the local on-site repulsion and the chemical potential are time-dependent, and in the presence of an external time-dependent electromagnetic field. We also evaluate these expressions for the homogeneous case in equilibrium, where all time dependence and external fields vanish. Unlike similar sum rules for the Fermi-Hubbard model, in the Bose-Hubbard model case, the sum rules often depend on expectation values that cannot be determined simply from parameters in the Hamiltonian like the interaction strength and chemical potential but require knowledge of equal-time many-body expectation values from some other source. We show how one can approximately evaluate these expectation values for the Mott-insulating phase in a systematic strong-coupling expansion in powers of the hopping divided by the interaction. We compare the exact moment relations to the calculated moments of spectral functions determined from a variety of different numerical approximations and use them to benchmark their accuracy.