We will consider the normal state of interacting Fermi particles without any longrange order. The normal state at low temperatures is called the Fermi liquid and is considered to be the system of free quasi-particles that is continuously connected with free Fermi gas. The concept of the Fermi liquid was introduced and developed by L. D. Landau. Landau's Fermi liquid theory, which concentrates rich contents into a simple theory, is a good example to solve many-body problems. In this chapter, we introduce the basis and main contents of the Fermi liquid.

Principle of continuity

The basis of the Fermi liquid theory is the principle of adiabatic continuity, which connects free Fermi gas with the Fermi liquid by introducing gradually an interaction among particles. There is a one-to-one correspondence between two states before and after the introduction of the interaction. States belonging to the same symmetry do not cross each other, and new states with the interaction can be represented by the old quantum number. Since the distribution of free Fermi gas is given by the Fermi distribution function n(k), that of the Fermi liquid is also written n(k). The state denoted by k, σ in the Fermi liquid is called the quasi-particle. For the system to be described by quasi-particles, the following condition is necessary.

Let us consider the basis of the Fermi liquid theory following the explanation of the Landau theory given by Nozières and Anderson.

P. W. Anderson has achieved many brilliant theories in the wide field of condensed matter physics. His book titled Basic Notions of the Condensed Matter Physics was published in 1984. In this book Anderson stresses two basic principles of condensed matter physics. One of the principles is ‘broken symmetry’. This means that condensed matter systems undergo phase transition to take a state possessing lower symmetry than that of the Hamiltonian. This statement corresponds to the appearance of a ferromagnetic state and a superconducting state, etc. at low temperatures. This principle manifests discontinuous change.

Another basic principle is the principle of ‘adiabatic continuity’. This principle tells us that when we study a generally complicated physical system we can refer to a simple system that contains the essential nature of the real system and understand the complicated system on the basis of knowledge of the simple system. Anderson stresses that the most beautiful and appropriate example showing the importance of the continuity principle is Landau's Fermi liquid theory. Following the continuity principle, we start from a non-interacting Fermi gas and introduce interactions among particles gradually. There exists a one-to-one correspondence between the free particle system before the introduction of the interactions and the Fermi liquid after the introduction. It is the basic character of the Fermi liquid at low temperatures that we can introduce interactions as slowly as possible owing to the long lifetime of quasi-particles.

Heavy fermion systems are explained on the basis of the Fermi liquid theory. The specific heat, magnetic susceptibility and electrical resistivity are discussed. Using Anderson's orthogonality theorem, we show that the Fermi liquid is nothing but a local spin singlet state at every site.

Heavy fermion systems

The Fermi liquid theory is independent of a model Hamiltonian, and can be applied to any system as long as the system remains a Fermi liquid. The theory tells us that even if the electron interaction becomes strong, physical quantities behave as those of the non-interacting Fermi gas. The difference between them with and without interaction is not qualitative but quantitative.

As seen at the Mott transition in Hubbard systems, the effective mass of the electron increases near the transition point. As a system realizing such a large effective mass, a Fermi liquid system called the heavy electron system or the heavy fermion system attracts general interest. The heavy fermion systems are composed of the rare earth metals such as Ce and Yb, and actinide atoms such as U. The heavy fermions are nothing but the quasi-particles in the Fermi liquid theory. As a quasi-particle in a strongly correlated electron system, the heavy fermion is an important issue to be studied in the development of the Fermi liquid theory. The heavy fermion realized in f -electron systems is one of the heavy quasi-particles appearing near the Mott transition.

The field of electronic structure is at a momentous stage, with rapid advances in basic theory, new algorithms, and computational methods. It is now feasible to determine many properties of materials directly from the fundamental equations for the electrons and to provide new insights into vital problems in physics, chemistry, and materials science. Increasingly, electronic structure calculations are becoming tools used by both experimentalists and theorists to understand characteristic properties of matter and to make specific predictions for real materials and experimentally observable phenomena. There is a need for coherent, instructive material that provides an introduction to the field and a resource describing the conceptual structure, the capabilities of the methods, limitations of current approaches, and challenges for the future.

The purpose of this and a second volume in progress is to provide a unified exposition of the basic theory and methods of electronic structure, together with instructive examples of practical computational methods and actual applications. The aim is to serve graduate students and scientists involved in research, to provide a text for courses on electronic structure, and to serve as supplementary material for courses on condensed matter physics and materials science. Many references are provided to original papers, pertinent reviews, and books that are widely available. Problems are included in each chapter to bring out salient points and to challenge the reader.

The printed material is complemented by expanded information available on-line at a site maintained by the Electronic Structure Group at the University of Illinois (see Ch. 24).

Summary

Definition and properties

Wannier functions [338, 759, 763] are orthonormal localized functions that span the same space as the eigenstates of a band or a group of bands. Extensive reviews of their properties have been given by Wannier [338], Blount [759], and Nenciu [339]. Here we consider properties relevant to understanding the electronic properties of materials and to presentday practical calculations.

The eigenstates of electrons in a crystal are extended throughout the crystal with each state having the same magnitude in each unit cell.

Summary

Augmented plane waves (APWs) and “muffin tins”

The augmented plane wave (APW) method, introduced by Slater [54] in 1937, expands the eigenstates of an independent-particle Schrödinger equation in terms of basis functions, each of which is represented differently in the two characteristic regions illustrated in Fig. 16.1. In the region around each atom the potential is similar to the potential of the atom and the solution for the wavefunction is represented in a form appropriate to the central region of an atom.