It is pointed out that the spectra of X-ray induced fast photoelectrons from metal should have a characteristic skew line shape resulting from Kondo-like many-electron interactions of the metallic conduction electrons with the accompanying deep hole in the final state. The same line shape should also occur for the discrete line spectra of X-rays emitted from metals. This mechanism could account for the well-known asymmetries observed for K alpha lines.

https://iopscience.iop.org/article/10.1088/0022-3719/3/2/010/pdf

Many-electron singularity in X-ray photoemission and X-ray line spectra from metals

Calculations of ground-state and excited-state properties of materials have been one of the major goals of condensed matter physics. Ground-state properties of solids have been extensively investigated for several decades within the standard density functional theory. Excited-state properties, on the other hand, were relatively unexplored in ab initio calculations until a decade ago. The most suitable approach up to now for studying excited-state properties of extended systems is the Green function method. To calculate the Green function one requires the self-energy operator which is non-local and energy dependent. In this article we describe the GW approximation which has turned out to be a fruitful approximation to the self-energy. The Green function theory, numerical methods for carrying out the self-energy calculations, simplified schemes, and applications to various systems are described. Self-consistency issue and new developments beyond the GW approximation are also discussed as well as the success and shortcomings of the GW approximation.

https://iopscience.iop.org/article/10.1088/0034-4885/61/3/002/pdf

The GW method

In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG method was later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method, including some guidelines for calculating physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi-liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean-field theory.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/NRG/BULLA=numerical_renormalization_group_method_for_quantum_impurity_systems.pdf

Numerical renormalization group method for quantum impurity systems

The Fermi gas model of one-dimensional conductors is reviewed. The exact solutions known for particular values of the coupling constants in a single chain problem (Tomonaga model, Luther-Emery model) are discussed. Renormalization group arguments are used to extend these solutions to arbitrary values of the couplings. The instabilities and possible ground states are studied by investigating the behaviour of the response functions. The relationship between this model and others is discussed and is used to obtain further information about the behaviour of the system. The model is generalized to a set of coupled chains to describe quasi-one-dimensional systems. The crossover from one-dimensional to three-dimensional behaviour and the type of ordering are discussed.

/pdf/the-fermi-gas-model-of-one-dimensional-conductors-4zpdg1dd6f.pdf

The Fermi gas model of one-dimensional conductors

Publisher Summary: This chapter discusses two major developments that have taken place over the past decade. First is the enormous wealth of energy band calculations that have had tremendous success in explaining the properties of specific solids, but in which the connection with first principles is not always apparent. Second is the spectacular progress of many-body theory applied to the solid state that has given a number of new results, although often of a rather general and formal nature, such as to provide the justification and a formal basis for a one-electron theory. The electron gas problem is treated in some detail here. The problem of the crystal potential is given due attention. It discusses the potential from the ion cores as well as from the valence electrons, and suggests schemes that incorporate essential exchange and correlation effects for the valence electrons. An energy band calculation that properly includes the effects of exchange and correlation describes the elementary excitations called quasi particles. Quasi-particle properties are usually discussed using the remarkable Landau theory of the Fermi liquid. This chapter gives a brief presentation of the theory and reviews the present status of calculations of the Fermi liquid parameters and how they are determined from experiments. (Less)

Effects of Electron-Electron and Electron-Phonon Interactions on the One-Electron States of Solids

The singularities of x-ray absorption or emission in metals are studied by a new "one-body" method, which describes the scattering of conduction electrons by the transient potential due to the deep hole. Using the linked-cluster theorem, the net transition rate in the time representation is expressed as the product of two factors: a one-electron transient Green's function $L$, and the deep-level Green's function $\mathcal{G}$. These factors obey simple Dyson equations, which can be solved asymptotically by using Muskhelishvili's method. The x-ray transition rate is found to behave as $\frac{1}{{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}}$, where $\ensuremath{\epsilon}$ is the frequency measured from the threshold, and $\ensuremath{\alpha}$ an exponent involving the various phase shifts ${\ensuremath{\delta}}_{l}$ which describe scattering by the deep hole. $\ensuremath{\alpha}$ may be g0 (infinite threshold) or 0 (zero threshold). The experimental implications of these results and their relation to the Friedel sum rule are briefly discussed.

Singularities in the X-Ray Absorption and Emission of Metals. III. One-Body Theory Exact Solution

X-ray emission and absorption spectra in metals may display singularities near the Fermi-level threshold. These singularities, predicted by Mahan, are due to final-state interactions between conduction electrons and the localized disturbance created by the x ray. The effect is treated within a simple model by the methods of perturbation theory. It is shown that even to lowest order one must sum the so-called parquet diagrams, in close analogy with the Abrikosov theory of the Kondo effect. Mahan's prediction is confirmed, and its validity discussed. Various secondary effects which could blur the singularity are analyzed.

Singularities in the X-Ray Absorption and Emission of Metals. I. First-Order Parquet Calculation

The many-body approach of the preceding paper is generalized to include self-energy and vertex renormalization is self-consistent fashion. The algebra of parquet diagrams is developed in terms of an unknown divergent parameter (related to the deep-hole Green's function $\mathcal{G}$, and of an unknown irreducible interaction $R$. The resulting interaction operator $\ensuremath{\gamma}$ is used to calculate $\mathcal{G}$ and $R$; the corresponding coupled equations are solved self-consistently in the weak coupling limit. The spectral densities of the deep-hole Green's function and of the x-ray response function are calculated explicitly; in the weak coupling limit, the Mahan singularity found in the preceding paper is unaffected by renormalization. The self-consistent formalism presented here describes the reaction of divergent fluctuations on themselves, and should, therefore, be useful in other more complicated problems, such as the Kondo effect.

Singularities in the X-Ray Absorption and Emission of Metals. II. Self-Consistent Treatment of Divergences

/pdf/optical-absorption-in-degenerate-semiconductors-27mgtkeoqy.pdf

P. Nozières

Papers

Singularities in the X-Ray Absorption and Emission of Metals. III. One-Body Theory Exact Solution

Singularities in the X-Ray Absorption and Emission of Metals. I. First-Order Parquet Calculation

Singularities in the X-Ray Absorption and Emission of Metals. II. Self-Consistent Treatment of Divergences

Optical absorption in degenerate semiconductors