Unraveling intrinsic correlation effects with angle-resolved photoemission spectroscopy.

Interaction effects can change materials properties in intriguing ways, and they have, in general, a huge impact on electronic spectra. In particular, satellites in photoemission spectra are pure many-body effects, and their study is of increasing interest in both experiment and theory. However, the intrinsic spectral function is only a part of a measured spectrum, and it is notoriously difficult to extract this information, even for simple metals. Our joint experimental and theoretical study of the prototypical simple metal aluminum demonstrates how intrinsic satellite spectra can be extracted from measured data using angular resolution in photoemission. A nondispersing satellite is detected and explained by electron-electron interactions and the thermal motion of the atoms. Additional nondispersing intensity comes from the inelastic scattering of the outgoing photoelectron. The ideal intrinsic spectral function, instead, has satellites that disperse both in energy and in shape. Theory and the information extracted from experiment describe these features with very good agreement.

Strategies to build functionals of the density, or functionals of Green's functions: what can we learn?

The many-body problem cannot in general be solved exactly, and one of the most prominent approximations is to build perturbation expansions. A huge variety of expansions are possible, which differ by the quantity to be expanded, the expansion variable, the starting point, and ideas of how to re-sum or terminate the series. Although much has been discussed and much has been done, some choices were made for historical reasons, in particular, limited computation or storage capacities. The present work aims to examine the justification for different choices made in different contexts, by comparing the components of functionals based on Green's functions on the one side, and on the charge density on the other side. Of particular interest will be the question of how to build an optimal starting point for the approximation of non-local quantities, making use of near- or far-sightedness, and daring to consider models beyond the homogeneous electron gas. This will include the use of connector approximations. We will also discuss why it is a good idea to build functionals of the density.

Design of auxiliary systems for spectroscopy.

The Kohn-Sham system is the prototypical example of an auxiliary system that targets, in principle exactly, an observable like the electronic density without the need to calculate the complicated many-body wavefunction. Although the Kohn-Sham system does not describe excited-state properties directly, it also represents a very successful strategy guideline for many spectroscopy applications. Here we propose a generalization of the Kohn-Sham idea. In many situations one is interested only in limited answers to specific questions, whereas in state-of-the-art approaches a lot of information is generally calculated that is not needed for the interpretation of experimental spectra. For example, when the target is a spectrum S(ω) like the optical absorption of a solid, within time-dependent density-functional theory (TDDFT) one calculates the whole response function χ(r,r',ω). Analogously, within many-body perturbation theory (MBPT) one calculates the whole one-particle Green's function G(r,r',ω), while only the total spectral function A(ω) is needed for angle-integrated photoemission spectra. In this contribution, we advocate the possibility of designing auxiliary systems with effective potentials or kernels that target only the specific spectral properties of interest and are simpler than the self-energy of MBPT or the exchange-correlation kernel of TDDFT. In particular, we discuss the fundamentals and prototypical applications of simplified effective kernels for optical absorption and spectral potentials for photoemission, and we discuss how to express these potentials or kernels as functionals of the density.

Insights into one-body density matrices using deep learning.

The one-body reduced density matrix (1-RDM) of a many-body system at zero temperature gives direct access to many observables, such as the charge density, kinetic energy and occupation numbers. It would be desirable to express it as a simple functional of the density or of other local observables, but to date satisfactory approximations have not yet been found. Deep learning is the state of the art approach to performing high dimensional regressions and classification tasks, and is becoming widely used in the condensed matter community to develop increasingly accurate density functionals. Autoencoders are deep learning models that perform efficient dimensionality reduction, allowing the distillation of data to the fundamental features needed to represent it. By training autoencoders on a large data-set of 1-RDMs from exactly solvable real-space model systems, and performing principal component analysis, the machine learns to what extent the data can be compressed and hence how it is constrained. We gain insight into these machine learned constraints and employ them to inform approximations to the 1-RDM as a functional of the charge density. We exploit known physical properties of the 1-RDM in the simplest possible cases to perform feature engineering, where we inform the structure of the models from known mathematical relations, allowing us to integrate existing understanding into the machine learning methods. By comparing various deep learning approaches we gain insight into what physical features of the density matrix are most amenable to machine learning, utilising both known and learned characteristics.

Nonlocal and Nonadiabatic Effects in the Charge-Density Response of Solids: A Time-Dependent Density-Functional Approach.

The charge-density response of extended materials is usually dominated by the collective oscillation of electrons, the plasmons. Beyond this feature, however, intriguing many-body effects are observed. They cannot be described by one of the most widely used approaches for the calculation of dielectric functions, which is time-dependent density functional theory (TDDFT) in the adiabatic local density approximation (ALDA). Here, we propose an approximation to the TDDFT exchange-correlation kernel which is nonadiabatic and nonlocal. It is extracted from correlated calculations in the homogeneous electron gas, where we have tabulated it for a wide range of wave vectors and frequencies. A simple mean density approximation allows one to use it in inhomogeneous materials where the density varies on a scale of 1.6 r_{s} or faster. This kernel contains effects that are completely absent in the ALDA; in particular, it correctly describes the double plasmon in the dynamic structure factor of sodium, and it shows the characteristic low-energy peak that appears in systems with low electronic density. It also leads to an overall quantitative improvement of spectra.

Exciton Band Structure in Two-Dimensional Materials.

Low-dimensional materials differ from their bulk counterparts in many respects. In particular, the screening of the Coulomb interaction is strongly reduced, which can have important consequences such as the significant increase of exciton binding energies. In bulk materials the binding energy is used as an indicator in optical spectra to distinguish different kinds of excitons, but this is not possible in low-dimensional materials, where the binding energy is large and comparable in size for excitons of very different localization. Here we demonstrate that the exciton band structure, which can be accessed experimentally, instead provides a powerful way to identify the exciton character. By comparing the ab initio solution of the many-body Bethe-Salpeter equation for graphane and single-layer hexagonal boron nitride, we draw a general picture of the exciton dispersion in two-dimensional materials, highlighting the different role played by the exchange electron-hole interaction and by the electronic band structure. Our interpretation is substantiated by a prediction for phosphorene.

Effects of low-energy excitations on spectral properties at higher binding energy: the metal-insulator transition of VO(2).

The effects of electron interaction on spectral properties can be understood in terms of coupling between excitations. In transition-metal oxides, the spectral function close to the Fermi level and low-energy excitations between d states have attracted particular attention. In this work we focus on photoemission spectra of vanadium dioxide over a wide (10 eV) range of binding energies. We show that there are clear signatures of the metal-insulator transition over the whole range due to a cross coupling of the delocalized s and p states with low-energy excitations between the localized d states. This coupling can be understood by advanced calculations based on many-body perturbation theory in the GW approximation. We also advocate the fact that tuning the photon energy up to the hard-x-ray range can help to distinguish fingerprints of correlation from pure band-structure effects.

Estimating excitonic effects in the absorption spectra of solids: problems and insight from a guided iteration scheme.

A major obstacle for computing optical spectra of solids is the lack of reliable approximations for capturing excitonic effects within time-dependent density functional theory. We show that the accurate prediction of strongly bound electron-hole pairs within this framework using simple approximations is still a challenge and that available promising results have to be revisited. Deriving a set of analytical formulas we analyze and explain the difficulties. We deduce an alternative approximation from an iterative scheme guided by previously available knowledge, significantly improving the description of exciton binding energies. Finally, we show how one can "read" exciton binding energies from spectra determined in the random phase approximation, without any further calculation.

Dynamical effects in electron spectroscopy.

One of the big challenges of theoretical condensed-matter physics is the description, understanding, and prediction of the effects of the Coulomb interaction on materials properties. In electronic spectra, the Coulomb interaction causes a renormalization of energies and change of spectral weight. Most importantly, it can lead to new structures, often called satellites. These can be linked to the coupling of excitations, also termed dynamical effects. State-of-the-art methods in the framework of many-body perturbation theory, in particular, the widely used GW approximation, often fail to describe satellite spectra. Instead, approaches based on a picture of electron-boson coupling such as the cumulant expansion are promising for the description of plasmon satellites. In this work, we give a unified derivation of the GW approximation and the cumulant expansion for the one-body Green's function. Using the example of bulk sodium, we compare the resulting spectral functions both in the valence and in the core region, and we discuss the dispersion of quasi-particles and satellites. We show that self-consistency is crucial to obtain meaningful results, in particular, at large binding energies. Very good agreement with experiment is obtained when the intrinsic spectral function is corrected for extrinsic and interference effects. Finally, we sketch how one can approach the problem in the case of the two-body Green's function, and we discuss the cancellation of various dynamical effects that occur in that case.

Total Energy beyond GW: Exact Results and Guidelines for Approximations.

The total energy and electron addition and removal spectra can, in principle, be obtained exactly from the one-body Green's function (GF). In practice, the GF is obtained from an approximate self-energy. In the framework of many-body perturbation theory, we derive different expressions that are based on an approximate self-energy, but that yield nevertheless, in principle, the exact exchange-correlation contribution to the total energy for any interaction strength. Response functions play a crucial role, which explains why, for example, ingredients from time-dependent density functional theory can be used to build these approximate self-energies. We show that the key requirement for obtaining exact results is the consistent combination of ingredients. Also when further approximations are made, as it is necessary in practice, this consistency remains the key to obtain good results. All findings are illustrated using the exactly solvable symmetric Hubbard dimer.

Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites.

The experimental valence band photoemission spectrum of semiconductors exhibits multiple satellites that cannot be described by the GW approximation for the self-energy in the framework of many-body perturbation theory. Taking silicon as a prototypical example, we compare experimental high energy photoemission spectra with GW calculations and analyze the origin of the GW failure. We then propose an approximation to the functional differential equation that determines the exact one-body Green's function, whose solution has an exponential form. This yields a calculated spectrum, including cross sections, secondary electrons, and an estimate for extrinsic and interference effects, in excellent agreement with experiment. Our result can be recast as a dynamical vertex correction beyond GW, giving hints for further developments.

Strong interplay between structure and electronic properties in CuIn(S,Se){2}: a first-principles study.

We present a first-principles study of the electronic properties of CuIn(S,Se){2} (CIS) using state-of-the-art self-consistent GW and hybrid functionals. The calculated band gap depends strongly on the anion displacement u, an internal structural parameter that measures lattice distortion. This contrasts with the observed stability of the band gap of CIS solar panels under operating conditions, where a relatively large dispersion of values for u occurs. We solve this apparent paradox considering the coupled effect on the band gap of copper vacancies and lattice distortions. The correct treatment of d electrons in these materials requires going beyond density functional theory, and GW self-consistency is critical to evaluate the quasiparticle gap and the valence band maximum.

Many-body perturbation theory and non-perturbative approaches: screened interaction as the key ingredient.

Many-body perturbation theory is often formulated in terms of an expansion in the dressed instead of the bare Green's function, and in the screened instead of the bare Coulomb interaction. However, screening can be calculated on different levels of approximation, and it is important to define what is the most appropriate choice. We explore this question by studying a zero-dimensional model (so called 'one-point model') that retains the structure of the full equations. We study both linear and non-linear response approximations to the screening. We find that an expansion in terms of the screening in the random phase approximation is the most promising way for an application in real systems. Moreover, by making use of the nonperturbative features of the Kadanoff-Baym equation for the one-body Green's function, we obtain an approximate solution in our model that is very promising, although its applicability to real systems has still to be explored.

Transforming nonlocality into a frequency dependence: a shortcut to spectroscopy.

Measurable spectra are often derived from contractions of many-body Green's functions. One calculates hence more information than needed. Here we present and illustrate an in principle exact approach to construct effective potentials and kernels for the direct calculation of electronic spectra. In particular, a dynamical but local and real potential yields the spectral function needed to describe photoemission. We discuss for model solids the frequency dependence of this "photoemission potential" stemming from the nonlocality of the corresponding self-energy.

Ab initio GW many-body effects in graphene.

We present an ab initio numerical many-body GW calculation of the band plot in freestanding graphene. We consider the full ionic and electronic structure introducing e-e interaction and correlation effects via a self-energy containing non-Hermitian and dynamical terms. With respect to the density-functional theory local-density approximation, the Fermi velocity is renormalized with an increase of 17%, in better agreement with the experiment. Close to the Dirac point the linear dispersion is modified by the presence of a kink, as observed by angle-resolved photoemission spectroscopy. We demonstrate that the kink is due to low-energy pi-->pi* single-particle excitations and to the pi plasmon. The GW self-energy does not open the band gap.

Anomalous angular dependence of the dynamic structure factor near Bragg reflections: graphite.

The electron energy-loss function of graphite is studied for momentum transfers q beyond the first Brillouin zone. We find that near Bragg reflections the spectra can change drastically for very small variations in q. The effect is investigated by means of first principle calculations in the random phase approximation and confirmed by inelastic x-ray scattering measurements of the dynamic structure factor S(q, omega). We demonstrate that this effect is governed by crystal local field effects and the stacking of graphite. It is traced back to a strong coupling between excitations at small and large momentum transfers.

Linear plasmon dispersion in single-wall carbon nanotubes and the collective excitation spectrum of graphene.

We have measured a strictly linear pi plasmon dispersion along the axis of individualized single-wall carbon nanotubes, which is completely different from plasmon dispersions of graphite or bundled single-wall carbon nanotubes. Comparative ab initio studies on graphene-based systems allow us to reproduce the different dispersions. This suggests that individualized nanotubes provide viable experimental access to collective electronic excitations of graphene, and it validates the use of graphene to understand electronic excitations of carbon nanotubes. In particular, the calculations reveal that local field effects cause a mixing of electronic transitions, including the "Dirac cone," resulting in the observed linear dispersion.