In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963).

The Matsubara Green function

The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble

In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H0 + Hi, where H0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part Hi in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “time” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.

This chapter provides an introduction to the theory of strongly disordered electron systems. As for the case of superconductivity, the understanding of transport and thermodynamical properties of dirty metals and semiconductors required the introduction of new concepts in solid-state physics.

In non-interacting systems the quantum interference effects yield corrections to the Drude–Boltzmann classical theory of electrical transport (developed in Chapter 11) and for sufficiently strong disorder the correction terms produce a transition to an insulating phase where the electrons are localized, named the Anderson localization (Anderson (1958)).

To emphasize the peculiarity of this transition, we recall that in a perfect crystal, according to Bloch theory, the electron states are extended plane waves modulated by the crystalline potential, grouped in finite energy bands. Within this picture, partial occupation of the bands leads to a metallic state, whereas insulators and semiconductors have an energy gap between the last fully occupied band and the first empty one. The classification of metallic and insulating states is done accordingly to the above filling procedure. The new aspect introduced by Anderson is the possibility of an insulating state arising from the localization of the electron wave function in a partially occupied band.

The effect of electron–electron interaction also introduces surprising effects. The resulting scenario is different from the single-particle picture and provides an effective Fermi liquid picture with frequency and momentum dependent Landau parameters. These parameters then become scale dependent and give rise to a complex flow of the renormalization group equations.

There exist several review articles which give an account of the problem from different viewpoints and at different stages of the historical development (Bergmann (1984); Lee and Ramakrishnan (1985); Altshuler and Aronov (1985); Finkelstein (1990); Belitz and Kirkpatrick (1994); Di Castro and Raimondi (2004)).

We will concentrate here on those aspects that we believe are fundamental for the understanding of the problem. Therefore we present, at first, the theory in a simple phenomenological way, then we introduce all the required technicalities necessary to proceed towards a microscopic theory.

D. Pines in his Editor's Foreword to the important series “Frontiers in Physics, a Set of Lectures” of the sixties and seventies of the past century (W. A. Benjamin, Inc.) was suggesting as a possible solution to “the problem of communicating in a coherent fashion the recent developments in the most exciting and active fields of physics” what he called “an informal monograph to connote the fact that it represents an intermediate step between lecture notes and formal monographs.”

Our aim in writing this book has been to provide a coherent presentation of different topics, emphasizing those concepts which underlie recent applications of statistical mechanics to condensed matter and many-body systems, both classical and quantum. Our goal has been indeed to reach an up to date version of the book Statistical Mechanics. A Set of Lectures by R. P. Feynman, one of the most important monographs of the series mentioned above. We felt, however, that it would have been impossible to give to a student the full flavor of the recent topics without putting them in the classical context as a continuous evolution. For this reason we introduced the basic concepts of thermodynamics and statistical mechanics. We have also concisely covered topics that typically can be found in advanced books on many-body theory, where usually the apparatus of quantum field theory is used.

In our book we have kept the technical apparatus at the level of the density matrix with the exception of the last four chapters. Up to Chapter 17 no particular prerequisite is needed except for standard courses in Classical and Quantum Mechanics. Chapter 18 provides an introduction to statistical quantum field theory, which is used in the last chapters. Chapters 20 and 21 cover topics which, although covered in recent monographs, are not commonly found in classical many-body books.

In our book then the student will find a bridge from thermodynamics and statistical mechanics towards advanced many-body theory and its applications. In our attempt to give a coherent account of several topics of condensed matter physics, we have at the same time preserved the personal point of view of the notes of our courses. Our bibliography is for this reason far from complete and the presentation of some topics is somewhat informal and partial. Many important contributions and fundamental references have been left out.