A quantum description is given of nanoskyrmions in two-dimensional textures with localized spins and itinerant electrons, isolated or coupled to leads, in or out of equilibrium. The spin-electron exchange is treated at the mean-field level, while tensor networks and exact diagonalization or nonequilibrium Green's functions are used for localized spins and itinerant electrons. We motivate our scheme via exact and mean-field benchmarks, then show by several examples that itinerant electrons distinctly affect the properties of quantum nanoskyrmions. Finally, we mention lines of future work and improvement of the approach.

It is known from density functional theory (DFT) calculations that RhSi has a multifold degenerate Dirac point at the Fermi energy, with the dominant states in the low-energy region displaying mostly Rh d character. Using DFT+U, we calculate the band structure by considering an effective local interaction on the Rh d states, with a realistic effective Hubbard U_eff=2.5 eV derived from a constrained random-phase approximation calculation, and find the emergence of a double hump structure close to the Fermi energy.By further deriving a low-energy tight-binding model from our first-principles results, we show that the double hump is a direct consequence of a competition between the Rh d-Rh d and Rh d-Si p interactions, which differ in their momentum dependence. As a consequence, through an artificial tuning of the energy level of the Si p orbitals this hump structure can be suppressed due to the effectively reduced Rh d -Si p interaction.This peculiar low-energy electronic structure additionally results in that a small hole/electron doping (∼ 0.1 %) can tune the Fermi surface topology, going from closed to open Fermi surfaces, which has dramatic consequences for the thermal transport.

The exchange-correlation potential formalism previously introduced and applied to the one-dimensional Hub-bard model has been extended to spin systems and applied to the case of the one-dimensional antiferromagnetic spin -21 Heisenberg model. Within the spin exchange-correlation potential formulation, a sum rule for spin systems is derived. The exchange-correlation potential for the Heisenberg model is extrapolated from exact diagonalization results of small antiferromagnetic Heisenberg clusters. This procedure is also employed to revisit and computationally improve the previous investigation of the exchange-correlation potential of the half-filled Hubbard model, which was based on the exchange-correlation potential of the dimer. Numerical comparisons with exact benchmark calculations for both the Heisenberg and the Hubbard models indicate that, starting from the exchange-correlation potential of a finite cluster, the extrapolation procedure yields a one-particle spectral function with favorable accuracy at a relatively low computational cost. In addition, a comparison between the ground-state energies for the one-dimensional Hubbard and Heisenberg models displays how the well-known similarity in behavior of the two models at large interactions manifests itself within the exchange-correlation potential formalism.

The exchange-correlation potential formalism previously introduced and applied to the one-dimensional Hubbard model has been extended to spin systems and applied to the case of the one-dimensional antiferromagnetic spin-1/2 Heisenberg model. Within the spin exchange-correlation potential formulation, a new sum rule for spin-systems is derived. The exchange-correlation potential for the Heisenberg model is extrapolated from exact diagonalization results of small antiferromagnetic Heisenberg clusters. This procedure is also employed to revisit and computationally improve the previous investigation of the exchange-correlation potential of the half-filled Hubbard model, which was based on the exchange-correlation potential of the dimer. Numerical comparisons with exact benchmark calculations for both the Heisenberg and the Hubbard models indicate that, starting from the exchange-correlation potential of a finite cluster, the extrapolation procedure yields a one-particle spectral function with favorable accuracy at a relatively low computational cost. In addition, a comparison between the ground state energies for the one-dimensional Hubbard and Heisenberg models displays how the well known similarity in behavior of the two models at large interactions manifests within the exchange-correlation potential formalism.

The self-screening error in the random-phase approximation (RPA) and the $GW$ approximation (GWA) is a well-known issue and has received attention in recent years with several methods for a correction being proposed. We here apply two of these, a self-screening and a so-called "self-polarization" correction scheme, to model calculations to examine their applicability. We also apply an explicit self-screening correction to \textit{ab-initio} calculations of real materials. We find indications for the self-polarization scheme to be the more appropriate choice of correction for localized states, and additionally we observe that it suffers from causality violations in the strongly correlated regime. The self-screening correction used in this work on the other hand significantly improves the description in more delocalized states. It provides a notable reduction in the remaining GWA error when calculating the band gaps of several semiconductors, indicating a physical explanation for a part of the remaining discrepancy in one-shot $GW$ compared to experiment, while leaving the localized semicore $d$ states mostly unaffected.

The exchange-correlation potential formalism previously introduced and applied to the one-dimensional Hubbard model has been extended to spin systems and applied to the case of the one-dimensional antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model. Within the spin exchange-correlation potential formulation, a sum rule for spin systems is derived. The exchange-correlation potential for the Heisenberg model is extrapolated from exact diagonalization results of small antiferromagnetic Heisenberg clusters. This procedure is also employed to revisit and computationally improve the previous investigation of the exchange-correlation potential of the half-filled Hubbard model, which was based on the exchange-correlation potential of the dimer. Numerical comparisons with exact benchmark calculations for both the Heisenberg and the Hubbard models indicate that, starting from the exchange-correlation potential of a finite cluster, the extrapolation procedure yields a one-particle spectral function with favorable accuracy at a relatively low computational cost. In addition, a comparison between the ground-state energies for the one-dimensional Hubbard and Heisenberg models displays how the well-known similarity in behavior of the two models at large interactions manifests itself within the exchange-correlation potential formalism.

Starting from the recently proposed dynamical exchange-correlation field framework, the equation of motion of the diagonal part of the many-electron Green function is derived, from which the spectral function can be obtained. The resulting equation of motion takes the form of the continuity equation of charge and current densities in electrodynamics with a source. An unknown quantity in this equation is the current density, corresponding to the kinetic energy. A procedure \`a la Kohn-Sham scheme is then proposed, in which the difference between the kinetic potential of the interacting system and the non-interacting Kohn-Sham system is shifted into the exchange-correlation field. The task of finding a good approximation for the exchange-correlation field should be greatly simplified since only the diagonal part is needed. A formal solution to the continuity equation provides an explicit expression for calculating the spectral function, given an approximate exchange-correlation field.

The exchange-correlation hole and potential of the homogeneous electron gas have been investigated within the random-phase approximation, employing the plasmon-pole approximation for the linear density response function. The angular dependence as well as the time dependence of the exchange-correlation hole are illustrated for a Wigner-Seitz radius $r_s=4$ (atomic unit). It is found that there is a substantial cancellation between exchange and correlation potentials in space and time, analogous to the cancellation of exchange and correlation self-energies. Analysis of the sum rule explains why it is more advantageous to use a non-interacting Green function than a renormalized one when calculating the response function within the random-phase approximation and consequently the self-energy within the well-established $GW$ approximation. The present study provides a starting point for more accurate and comprehensive calculations of the exchange-correlation hole and potential of the electron gas with the aim of constructing a model based on the local density approximation as in density functional theory.

Downfolding Methods in Many-Electron Theory is a comprehensive description of the last 30 years of study in this field, starting from LDA+U to LDA+DMFT and arriving at the recent work on multitier self-consistent GW+EDMFT. It focuses on different levels of first-principles electronic structure theories for strongly correlated electrons and outlines the downfolding method as a general approach—proven to be very productive in first-principles electronic structure of materials with strong electron correlations. This important book:Summarizes the latest developments of electronic structure methods for correlated materialsDescribes each key method from the simplest approximation to the latest developmentsPresents specific topics not often addressed in existing literature Downfolding Methods in Many-Electron Theory is an invaluable resource for researchers and practitioners working in electronic structure in condensed matter physics. The book presents the topic at a level also appropriate for graduate students.

This chapter describes the theoretical framework underlying the GW approximation for the self-energy and the associated random-phase approximation for the linear density response function. After a summary of the success of the GWA in greatly improving the LDA quasiparticle bandgaps and dispersions, emphasis is then shifted to the fundamental difficulties of the GWA as a first-order perturbation theory in the screened interaction in dealing with long-range collective charge excitations (plasmons) and strongly correlated systems. Some examples are presented as illustrations and the difficulties are analyzed using simple models to illustrate the problems. Attempts to go beyond the GWA using diagrammatic techniques are described, leading to the inevitable conclusion that for strongly correlated systems, a new non-perturbative method is required.

The magnon-phonon interaction is receiving growing attention due to its key role in spin caloritronics and the emerging field of acoustic spintronics. At resonance, this magnetoelastic interaction forms magnon polarons, which underpin exotic phenomena such as magnonic heat currents and phononic spin, but is mostly investigated using mesoscopic spin-lattice models. Motivated to integrate the magnon-phonon interaction into first-principles many-body electronic structure theory, we set out to derive the exchange contribution, which is subtler than the spin-orbit contribution, using Schwinger functional derivatives. To avoid having to solve the famous Hedin-Baym equations self-consistently, the phonons are treated as perturbations to the electronic structure. A formalism based on imposing a crossing-symmetric electron-electron interaction is developed in order to treat charge and spin on equal footing to respect the Pauli exclusion principle. Due to spin conservation, the magnon-phonon interaction first enters to second order through the magnon-magnon interaction, which renormalizes the magnons. We show by iteration that the magnon-magnon interaction contains a "screened T matrix" term and an arguably more important term which, in the local-spin limit, enables first-principles phonon emission and absorption amplitudes, predicted by phenomenological magnetoelastic models. These terms are, respectively, of first and second order in the screened collective four-point interaction 1/V-a crossing-symmetric analog of Hedin's W. Proof-of-principle results are presented at varying temperatures for an isotropic magnon spectrum in three dimensions in the presence of a flat optical phonon branch.

The spectral functions of the one-band half-filled 1D Hubbard chain are calculated using the exchange-correlation potential formalism developed recently. The exchange-correlation potential is adopted from the exact potential derived from the Hubbard dimer. Within an approximation in which the full Green function is replaced by a non-interacting one, the spectral functions can be calculated analytically. Despite the simplicity of the approximation, the resulting spectra are in favorable agreement with the more accurate results obtained from the dynamic density-matrix renormalization group method. In particular, the calculated band gap as a function of $U$ is in close agreement with the exact gap obtained from the Bethe ansatz. In addition, the formal general solution to the equation of motion of the Green function is presented and the difference between the traditional self-energy approach and the exchange-correlation potential formalism is also discussed and elaborated. A simplified Holstein Hamiltonian is considered to further illustrate the general form of the exchange-correlation potential.

The magnon Hedin's equations are derived via the Schwinger functional derivative technique, and the resulting self-consistent Green's function method is used to calculate ground state spin patterns and magnetic structure factors for 2-dimensional magnetic systems with frustrated spin-1/2 Heisenberg exchange coupling. Compared to random-phase approximation treatments, the inclusion of a self-energy correction improves the accuracy in the case of scalar product interactions, as shown by comparisons between our method and exact benchmarks in homogeneous and inhomogeneous finite systems. We also find that for cross-product interactions (e.g. antisymmetric exchange), the method does not perform equally well, and an inclusion of higher corrections is in order. Aside from indications for future work, our results clearly indicate that the Green's function method in the form proposed here already shows potential advantages in the description of systems with a large number of atoms as well as long-range interactions.