Periodic Reporting for period 1 - COUPE (COUPE - COUPling Excitations)
Période du rapport: 2020-05-01 au 2022-04-30
Although the general theory of photoemission is understood since many years, we do not fully understand each and every phenomenon observed in the experiment and increased resolution and precise control over radiation sources have led to numerous new discoveries of previously inaccessible spectroscopic signatures, often linked to complex excitation processes. The main challenge in their interpretation to both theoreticians and experimentalists alike is that materials contain a very large number of electrons interacting via the Coulomb-interaction not yielding to a simple or intuitive interpretation. Instead, complex excitation processes, where excitations are coupled through interactions, govern the physics of the phenomena. A striking signature for this is that spectra often consist of one or more dominant peaks (called quasi-particle), followed by a series of secondary structures (called satellites) at lower energies. The latter, which can be very strong, are pure interaction effects and may be interpreted in terms of coupling of a quasiparticle to a bosonic excitation of the material: a plasmon, exciton, phonon, or any other boson. To solve the electron-boson problem, and, first of all, to set up the description in terms of the most pertinent quasiparticles and bosons, is a major challenge of condensed matter theory. In the COUPE project (derived from COUPling Excitations) we strive to tackle this problem by bridging two existing theories. Today, quasiparticles are most often described in the GW approximation to many-body perturbation theory, whereas for bosonic excitations one can resort to time-dependent density-functional theory (TDDFT). Only a few approaches exist to describe the coupled problem, with severe limitations. For example, a cumulant expansion on top of GW describes plasmon satellites, but cannot couple to spin excitations, and there is yet no systematic way for improvement. COUPE proposed a new combination of GW and TDDFT which couples quasiparticles to all kinds of bosonic excitations. This combination relies on a recently developed “connector theory” (CT), which allows using results from the homogeneous electron gas to describe inhomogeneous materials.
The COUPE project is based on the Connector Theory (CT) approach and the non-local, dynamic exchange-correlation kernel “2p2h” for the homogeneous electron gas (HEG). In the initial project phase we analysed the analytic properties of this kernel and studied it for a density similar to the average density of a very similar (but real) material: sodium. We found an analytic way to represent the (only numerically available) data for the HEG and studied the effect of inclusion of the analytic form on a very reduced set of equations of the GWΓ approach. This initial study confirmed the suspicion of the dynamic (frequency-dependent) nature of the “2p2h”-kernel being indeed responsible for the double-plasmon feature in inelastic x-ray scattering but also for the position and amplitude of the satellites in photoemission, thus improving the existing theories with respect to the experimental reference. This also confirmed the suitability of the average density as a possible connector to make the connection between the homogeneous electron gas and inhomogeneous real systems. Given the very promising results of the model study we re-implemented the “2p2h”-kernel in the most recent version of the TDDFT-code dp to be available for every material, which involved first to reproduce the previous results of Panholzer et al. who investigated the dynamic structure factor of sodium. Then, the same code was modified to produce the required input to the GWΓ equations in the abinit program. To achieve this the tabulated kernel (which is only available in a narrow window of reciprocal space and time) was interpolated and extrapolated according to known limits and transformed in an analytic form to allow for analytic continuation. All these changes were continuously integrated in the master branch of the dp-code, test cases written and reference output produced. This newly implemented code was then applied to the simulation of photoemission spectra of sodium and silicon. Because the effects were smaller than expected a follow-up study aimed at enhancing the “2p2h” kernel.
We were able to show by our model study that using a modified screened Coulomb interaction in the GWΓ approximation via a combination with TDDFT is indeed a viable route to access previously unavailable spectral features in photoemission spectra. On the TDDFT-side we extended the capabilites of the dp software to use three new kernels (adiabatic mean density approximation, the Corradini-kernel, and the “2p2h”-kernel) for an arbitrary electronic density (any kind of material) to not only calculate inelastic x-ray scattering, but also to create the required test-electron screening data to connect it with the GWΓ approximation in the abinit code by devising an analytic representation of the tabulated data allowing not only to work at arbitrary values outside of the tabulated area, but also to permit the analytic continuation to the complex frequency domain. It is expected that the availability of this new implementation allows for many follow-up discoveries not only in the context of the combined GWΓ/TDDFT approach sought in COUPE but also in the general application of the dp software to the simulation of materials.