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arXiv:cond-mat/9811356 · Physical review. B, Condensed matter

Evolution of one-particle and double-occupied Green functions for the Hubbard model, with interaction, at half-filling with lifetime effects within the moment approach

Sandra Schafroth, José Jesús Rodríguez-Núñez

Published Aug 15, 1999• 0 citations
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Abstract

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 Matter 10, L391 (1998); J. J. Rodr\'{\i}guez-N\'u\~nez, S. Schafroth, and H. Beck, Physica B (to be published); (unpublished)].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.

DOI
10.1103/physrevb.60.5366

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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.
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 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.
Restudying the non-Fermi-liquid one-particle Green functions (NFLGF) we have extended the work of Balatsky [Philos. Mag. Lett. 68, 251 (1993)] and Yin and Chakravarty [Int. J. Mod. Phys. B 10, 805 (1996)], among others. We use the moment approach of Nolting [Z. Phys. 255, 25 (1972)] to compute the unknown parameters of the NFLGF's in the framework of the Hubbard model. The zeroth-order moment requires that our one-particle Green functions describe fermionic degrees of freedom. In order to satisfy the first-order sum rule, a renormalization, $\ensuremath{\gamma}\ensuremath{\ne}1,$ of the free-electron mass is called for. The second-order sum rule or moment imposes a relation between the non-Fermi-liquid parameter, $\ensuremath{\alpha},$ the Coulomb interaction, U, and the frequency cutoff, ${\ensuremath{\omega}}_{c}.$ We have calculated the effect of the mass renormalization factor, $\ensuremath{\gamma},$ on some physical quantities, such as (i) the correlated momentum distribution function, ${n}_{c}(\stackrel{\ensuremath{\rightarrow}}{k}),$ close to the effective chemical potential, at T=0;T=0;T=0; (ii) the superconducting critical temperature, Tc;{T}_{c};Tc​; and (iii) the superconducting critical interaction, ${\ensuremath{\lambda}}_{\mathrm{cr}},$ and compared them with analytical results found in the literature. Also, we have calculated the isotope effect, ${\ensuremath{\alpha}}^{\ensuremath{'}},$ for non-Fermi-liquid systems, which reduces to ${\ensuremath{\alpha}}^{\ensuremath{'}}=1/2$ (the BCS result) when $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}0.$ As a case of non-Fermi-liquid systems, in the Appendix, we have studied two inequivalent coupled Hubbard layers for which we calculate the one-particle spectral functions on the layers and perpendicular to them. We discuss the features which appear due to the shift in the two effective chemical potentials and propose some experiments to detect the features found from our expressions.