The analytic continuation of the GW self-energy from the imaginary to the real-energy axis is a central difficulty for approaches exploiting the favorable properties of response functions at imaginary frequencies. Within a scheme merging contour-deformation and analytic-continuation techniques, we show on the basis of extensive calculations for large molecular sets that it is preferable to perform an analytic continuation of the dynamically screened-Coulomb potential W rather than the much more structured self-energy operator. The case of states lying far away from the gap, including core states, is addressed by generalizing the analytic-continuation scheme, accounting further for quasiparticle lifetimes.
The analytic continuation of the GW self-energy from the imaginary to the real-energy axis is a central difficulty for approaches exploiting the favorable properties of response functions at imaginary frequencies. Within a scheme merging contour-deformation and analytic-continuation techniques, we show on the basis of extensive calculations for large molecular sets that it is preferable to perform an analytic continuation of the dynamically screened-Coulomb potential W rather than the much more structured self-energy operator. The case of states lying far away from the gap, including core states, is addressed by generalizing the analytic-continuation scheme, accounting further for quasiparticle lifetimes.
The analytic continuation of the GW self-energy from the imaginary to the real energy axis is a central difficulty for approaches exploiting the favourable properties of response functions at imaginary frequencies. Within a scheme merging contour-deformation and analytic-continuation techniques, we show on the basis of extensive calculations for large molecular sets that it is preferable to perform an analytic continuation of the dynamically screened Coulomb potential W rather than the much more structured self-energy operator. The case of states lying far away from the gap, including core states, is addressed by generalizing the analytic continuation scheme, accounting further for quasiparticle lifetimes.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
We present a new method for efficient, accurate calculations of many-body properties of periodic systems. The main features are (i) use of a real-space/imaginary-time representation, (ii) avoidance of any model form for the screened interaction W, (iii) exact separation of W and the self-energy $\ensuremath{\Sigma}$ into short- and long-ranged parts, and (iv) the use of novel analytical continuation techniques in the energy domain. The computer time scales approximately linearly with system size. We give results for jellium and silicon, including the spectral function of silicon obtained from the Dyson equation.
The self-energy Pad\'e approach is discussed as an analytic continuation of numerical methods that evaluate correlation functions in imaginary time. Instead of direct analytic continuation of the correlation functions, the Pad\'e method is applied for the self-energy in this approach, which is then used for deriving the Green's functions at real energies. We demonstrate that the self-energy Pad\'e approach is more stable and robust against statistical errors compared to the direct way. The characteristics and success of the self-energy Pad\'e approach are analyzed by actual calculations for the illustrative examples of the noninteracting Anderson lattice and interacting Hubbard model. A zero-gap Kondo lattice model with linearly vanishing conduction electron density of states at the Fermi level is also studied by using the self-energy Pad\'e approach as an analytic continuation. We investigate the properties of the Kondo insulating state including the dependence of the insulating gap on the Kondo coupling and coherence effects. Furthermore, we identify two energy scales from dynamic and thermodynamic quantities that are associated as a direct and an indirect gap in a band hybridization picture.
The GW formalism, a Green’s function many-body perturbation theory, is growing in popularity for the description of the electronic properties of condensed matter systems in solid-state physics, and more recently chemistry. Unfortunately, its application to complex systems of interest in nanosciences, chemistry, or even biology, is hampered by the large associated computing cost, in particular in the case of disordered systems, or systems immersed in an opened environment (a solvent, a molecular medium, an electrode, etc.) The goal of the present PhD thesis is to focus on the development of multiscale techniques, merging high-level many-body treatments of the subsystem of interest, with a simplified but fully ab initio description of the electrostatic and dielectric environment. Such approaches aim to go beyond classical parametrized models, mainly developed in the quantum chemistry community, which are based on a continuum (“polarizable continuum model”) or discrete (QM/MM) description of the environment.To reach such a goal, we adopt a divide-and-conquer fragmentation scheme for the environment, particularly suited to molecular systems. This leads to a block-diagonal non- interacting electron susceptibility, decreasing the algorithmic complexity from quartic to cubic. To reduce the prefactor associated with the inversion of the Dyson equation for the screened Coulomb potential W, we have further developed a compression algorithm for the susceptibility operator. The automatic computation of an extremely compact polarization basis set allows a large reduction of the size of the susceptibility blocks, associated to the fragments in the environment. Such a method enables us to compute the dielectric response of systems made of several hundred thousand atoms, with an excellent accuracy when it comes to reproduce the effect of the environment as a response to an excitation in the immersed subsystem. This approach is presented through the study of fullerene bulk, surface and subsurface crystals.While the GW formalism is dynamical, with a frequency-dependent screened Coulomb potential W, a first study is done adopting a static approximation (low-frequency limit) for the screening properties of the environment. Such an approach follows the traditional semi-empirical models of a polarizable environment. This PhD thesis assesses the validity of such an approximation, which assumes an instantaneous response (adiabatic limit) of the environment to an electronic excitation, thanks to an explicit comparison with a fully dynamical dielectric response of the environment. The study of a surface of fullerenes, as well as a water molecule inside a metallic carbon nanotube, show that a static description of the environment leads to errors on the polarization energy below 10%, provided that the “folding” of the environment is treated in a proper way.The fragment approach is also applied to covalent insulator crystals, and more particularly to hexagonal boron nitride (h-BN). We explain how to compute the energy levels of point defects in h-BN, in the true dilute limit, and we give the asymptotic scaling laws for the renormalization of these energy levels, from the monolayer to a (n)-layer system. This study highlights thus the possibility to apply the fragment approach to covalent insulator systems, a possibility hinging probably on the short range behavior of the susceptibility in these systems.All of these developments, extending ab initio many-body methods to increasingly complex systems, have been implemented in the massively parallel code beDeft, dedicated to the study of the electronic properties of large scale systems.
We develop a reliable parameter-free analytic continuation method for quantum many-body calculations. Our method is based on a kernel grid, a causal spline, a regularization using the second-derivative roughness penalty, and the L-curve criterion. We also develop the L-curve averaged deviation to estimate the precision of our analytic continuation. To deal with statistically obtained data more efficiently, we further develop a bootstrap-averaged analytic continuation method. In the test using the exact imaginary-frequency Green's function with added statistical error, our method produces the spectral function that converges systematically to the exact one as the statistical error decreases. As an application, we simulate the two-orbital Hubbard model for various electron numbers with the dynamical-mean field theory in the imaginary time and obtain the real-frequency self-energy with our analytic continuation method, clearly identifying a non-Fermi-liquid behavior as the electron number approaches the half filling from the quarter filling. Our analytic continuation can be used widely, and it will facilitate drawing clear conclusions from imaginary-time quantum many-body calculations.