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Physica B Condensed Matter

Evolution of single-particle green functions with correlation in the hubbard model at half-filling: sum rules for the spectral density

José Jesús Rodríguez-Núñez, Sandra Schafroth, H Beck

Published Jan 1, 1999• 0 citations
Publisher
DOI
10.1016/s0921-4526(98)00940-5

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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

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 Matter 10, L391 (1998); J. J. Rodr\'{\i}guez-N\'u\~nez, S. Schafroth, and H. Beck, Physica B (to be published); (unpublished)].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

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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.
One of the challenges in many-body physics is determining the effects of phonons on strongly correlated electrons. The difficulty arises from strong correlations at differing energy scales -- for band metals, Migdal-Eliashberg theory accurately determines electron-phonon coupling effects due to the absence of vertex corrections -- but strongly correlated electrons require a more complex description and the standard Migdal-Eliashberg approach does not necessarily apply. In this work, we solve for the atomic limit Green's function of the Holstein-Hubbard model with both time-dependent electron-electron and electron-phonon couplings. We derive nonequilibrium spectral moment sum rules for the first four moments of the single-particle Green's function. We then analytically confirm that this Green's function exactly satisfies the zeroth, first, and second moment sum rules, and provide numerical evidence that the third spectral moment sum rule also holds. Finally we calculate the two-particle Green's function in the atomic limit. These two functions are required inputs to create a self-consistent strong-coupling expansion for the lattice Green's functions of the Hubbard-Holstein model.
We derive exact operator average expressions for the first two spectral moments of nonequilibrium Green's functions for the Falicov-Kimball model and the Hubbard model in the presence of a spatially uniform, time-dependent electric field. The moments are similar to the well-known moments in equilibrium, but we extend those results to systems in arbitrary time-dependent electric fields. Moment sum rules can be employed to estimate the accuracy of numerical calculations; we compare our theoretical results to numerical calculations for the nonequilibrium dynamical mean-field theory solution of the Falicov-Kimball model at half-filling.