Participants: Matteo Bertocchi, Silke Biermann, Fabio Caruso, Pierluigi Cudazzo, Rex Godby, Hardy Gross, Matteo Guzzo, Ralf Hambach, Linda Hung, Giovanna Lani, John Rehr, Lucia Reining, Pina Romaniello, Claudia Rödl, Angel Rubio, Andreas Schroen, Nader Slama, Francesco Sottile, Lorenzo Sponza
Discussion:
MONDAY 11-4-2011:
Fabio Caruso: self-consistent GW
The Dyson's equation has been solved self-consistently in the GW0 (with the screened interaction W0 fixed at the RPA level of screening) and GW approximation for the self-energy of a set of molecules. By expanding the frequency dependence of the Green's function in a basis of poles, it is possible to perform calculations with only a few tens of grid points, with a computational speed up of more than an order of magnitude compared to real-frequency approaches. With respect to G0W0 calculation based DFT-LDA, the quasi-particle (QP) peaks in the spectral function are improved at self-consistency, and provide ionization energies in better agreement with experiments. Small sodium clusters were chosen as test-cases to study the effects self-consistency on plasmon satellites. Nonetheless, working in imaginary frequency we do not have enough resolution to resolve secondary structures in the spectral function, such as plasmarons and satellites. The analytic continuation -necessary to obtain the spectral function in real frequency- has indeed a limited number of degrees of freedom, which may not capture all the features of the self-energy, eventually leading to unphysical features in the spectrum.
Giovanna Lani: self-consistent procedure and starting point
We have investigated certain flavours of the GW and GWΓ approximations to the self-energy within a model framework, so called 1-point model. In such framework one deals with algebraic equations and the Green's function (G) can be obtained from the exact solution of a linearized version of the functional differential equation [L.P. Kadanoff and G. Baym, Quantum Statistical Mechanics, W.A. Benjamin Inc., New York, 1964] for the one-body G. We find that for small interaction strength, self-consistent GW is a very good approximation (wrt i.e. to the one-shot calculation) to the exact solution, although a first order Γ, employed in a truly sc-scheme, gives an even better result. When a wider range of interaction is examined, the sc GW and the sc GWΓ become comparable, and for very large interaction it even seems that the simpler sc GW scheme is the best approximation; while a G0W0 + Γ dramatically diverges from the exact solution. Going beyond G0W0 has hence advantages. However, it requires the solution of higher order equations for G (for example in the sc $GW$ a 2nd order equation for G and hence two roots are obtained) and somewhat a criteria has to be chosen to pick the physical solution. Furthermore we find that, employing the "classical" iterative scheme for a sc GW, i.e. starting from the Dyson equation G=G0+G0ΣG, no matter what starting point, one always converges to the physical solution. A different iterative scheme, instead (like starting from Σ=G0-1-G-1), may lead to an unphysical solution.
John Rehr: charge-transfer satellites
Satellites from core-excitations in strongly correlated d- (or f-electron) systems exhibit strong "charge-transfer satellites" in the spectra which are different in character from those due to plasmon excitations. This effect is due to the strong coulomb interaction between the core-hole and localized states, and can be illustrated using a 3-level model, as discussed by Lee, Gunnarsson and Hedin (LGH), J. Phys. CM, R489 (1999). When the core-hole is created a localized d-state is pulled below the "ligand" states on neighboring atoms. This gives rise to an XPS spectrum with two peaks. In the leading peak an electron from the ligand orbital is transferred to the absorbing atom while for the satellite at a few eV there is no charge transfer. The model can be treated using a Hubbard-like prescription and solved exactly or approximated in various ways, as discussed by LGH. Although the model is relatively crude, it gives good results for the strong 6 eV satellites observed in the Hi Tc cuprates LSCO and NCCO.
Matteo Guzzo: Cumulant and satellites
The GW method from Many-Body Perturbation Theory (MBPT) has shown to be very successful in the description of photoemission spectra in a variety of systems. In particular, GW is known to give good quasiparticle properties like band-gaps. However, it has shown some limitations regarding the description of other more complex spectral features like satellites. Satellite peaks in photoemission come from higher-order excitations, e.g. plasmons, and are still poorly studied in real materials. In one-shot GW the spectral function $A(\omega)$ can describe additional features beside the quasiparticle peaks, but these satellites are known to be too weak and too low in energy, as it appears from calculations on the Homogeneous Electron Gas (L. Hedin et al., J. Res. Natl. Bur. Stand. Sect. A 74A, 417 (1970)) and some real materials (F. Aryasetiawan et al., Phys. Rev. Lett. 77 (1996) 2268; A.S. Kheifets et al. Phys. Rev. B 68 (2003) 233205). It is not clear wether self-consistency could improve the results. We have thoroughly studied the one-shot GW spectral function of bulk Si as a test case, trying to understand the flaws of GW in this respect. Here we compare the theory with new XPS experimental data. These data clearly show multiple plasmon replicas in the photoemission spectrum which are not reproducible within GW . The cumulant expansion approximation (CE) for the Green’s function, very popular in core-level spectroscopies, has been able to describe satellites correctly within this framework. In the case of valence spectroscopy much less data are available and results are not conclusive . We obtain the cumulant expansion by means of approximations on the 1-particle Green’s function’s equation of motion, giving a rigorous derivation for this approximation. We show that the CE gives excellent agreement with experiment and discuss how to improve this result even further.
Rex Godby: vertex corrections from DFT(DFT-like)-based approaches Some results ["Vertex corrections in localized and extended systems", Andrew J. Morris, Martin Stankovski, Kris T. Delaney, Patrick Rinke, P. García-González and R.W. Godby, Physical Review B 76 155106 (2007); Martin Stankovski et al to be published] on density-based approaches to calculating the vertex function to go beyond the GW level are reported. The most promising approach uses an averaged-density exchange-correlation potential in the vertex, and yields a vertex that is properly non-local. There were indications that the goal of a workable self-energy approximation that gives good spectral *and* total-energy properties was brought closer.
TUESDAY 12-4-2011:
Andreas Schrön: The Ground State of CoO in Different DFT+U Approaches The determination of the ground state of CoO strongly depends on a correct description of the Co 3d-states. Common to all 3d-transition metal oxides is the improper treatment of the on-site Coulomb interaction for the 3d-states of the transition metal atoms within standard DFT approximations like LDA and GGA. This failure can be cured by a set of similar correction methods generally named DFT+U. In the special cases of CoO and FeO a non-vanishing orbital moment is measured experimentally and thus spin-orbit-coupling is expected to have considerable influence on description of the ground state as well. We introduce the DFT+U approach of Lichtenstein, which depends upon the two parameters U and J and relate it to Dudarev's DFT+U approach. The latter is shown to be equivalent to Lichtenstein's approach with vanishing J. Results from ground-state calculations of CoO within the LDA and GGA approximations to DFT together with Lichtensteins DFT+U correction are presented. We stress especially the problem to obtain the correct orbital momentum within this approach and discuss the dependence of the results upon the J parameter.