This textbook contains a pedagogical introduction to the theory of Green's functions in and out of equilibrium, and is accessible to students with a standard background in basic quantum mechanics and complex analysis. Two main motivations prompted us to write a monograph for beginners on this topic.

The first motivation is research oriented. With the advent of nanoscale physics and ultrafast lasers it became possible to probe the correlation between particles in excited quantum states. New fields of research like, e.g., molecular transport, nanoelectronics, Josephson nanojunctions, attosecond physics, nonequilibrium phase transitions, ultracold atomic gases in optical traps, optimal control theory, kinetics of Bose condensates, quantum computation, etc. added to the already existing fields in mesoscopic physics and nuclear physics. The Green's function method is probably one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has already proven to be extremely useful in several of the aforementioned contexts. Extending the method to deal with the new emerging nonequilibrium phenomena holds promise to facilitate and quicken our comprehension of the excited state properties of matter. At present, unfortunately, to learn the nonequilibrium Green's function formalism requires more effort than learning the equilibrium (zero-temperature or Matsubara) formalism, despite the fact that nonequilibrium Green's functions are not more difficult. This brings us to the second motivation.

In this chapter I discuss the problem of electrons moving on a plane in the presence of an external uniform magnetic field perpendicular to the system. This is a subject of great interest from the point of view of both theory and experiment. The explanation of the remarkable quantization of the Hall conductance observed in MOSFETs and in heterostructures has demanded a great deal of theoretical sophistication. Concepts drawn from branches of mathematics, such as topology and differential geometry, have become essential to the understanding of this phenomenon. In this chapter I will consider only the quantum Hall effect in non-interacting systems. This is the theory of the integer Hall effect. The fractional quantum Hall effect (FQHE) is discussed in Chapter 13. The related subject of topological insulators is discussed in Chapter 16.

The chapter begins with a description of the one-electron states, both in the continuum and on a 2D lattice, followed by a summary of the observed phenomenology of the quantum Hall effect. A brief discussion of linear-response theory is also presented. The rest of the chapter is devoted to the problem of topological quantization of the Hall conductance.

One-dimensional Fermi systems

We will now consider the case of one-dimensional (1D) Fermi systems for which the Landau theory fails. The way it fails is quite instructive since it reveals that in one dimension these systems are generally at a (quantum) critical point, and it will also teach us valuable lessons on quantum criticality. It will also turn out that the problem of 1D Fermi systems is closely related to the problem of quantum spin chains. This is a problem that has been discussed extensively by many authors, and there are several excellent reviews on the subject (Emery 1979; Haldane, 1981). Here I follow in some detail the discussion and notation of Carlson and coworkers (Carlson et al., 2004).

One-dimensional (and quasi-1D) systems of fermions occur in several experimentally accessible systems. The simplest one to visualize is a quantum wire. A quantum wire is a system of electrons in a semiconductor, typically a GaAs–AlAs heterostructure built by molecular-beam epitaxy (MBE), in which the electronic motion is laterally confined along two directions, but not along the third. An example of such a channel of length L and width d (here shown as a two-dimensional (2D) system) is seen in Fig. 6.1. Systems of this type can be made with a high degree of purity with very long (elastic) mean free paths, often tens of micrometers or even longer. The resulting electronic system is a 1D electron gas (1DEG). In addition to quantum wires, 1DEGs also arise naturally in carbon nanotubes, where they are typically multi-component (with the number of components being determined by the diameter of the nanotube).

Field theory and condensed matter physics

Condensed matter physics is a very rich and diverse field. If we are to define it as being “whatever gets published in the condensed matter section of a physics journal,” we would conclude that it ranges from problems typical of material science to subjects as fundamental as particle physics and cosmology. Because of its diversity, it is sometimes hard to figure out where the field is going, particularly if you do not work in this field. Unfortunately, this is the case for people who have to make decisions about funding, grants, tenure, and other unpleasant aspects in the life of a physicist. They have a hard time figuring out where to put this subject which is neither applied science nor dealing with the smallest length scales or the highest energies. However, the richness of the field comes precisely from its diversity.

The past few decades have witnessed the development of two areas of condensed matter physics that best illustrate the strengths of this field: critical phenomena and the quantum Hall effect. In both cases, it was the ability to produce extremely pure samples which allowed the discovery and experimental study of the phenomenon. Their physical explanation required the use of new concepts and the development of new theoretical tools, such as the renormalization group, conformal invariance, and fractional statistics.

Preface to the first edition

This volume is an outgrowth of the course “Physics of Strongly Correlated Systems” which I taught at the University of Illinois at Urbana-Champaign during the Fall of 1989. The goal of my course was to present the field-theoretic picture of the most interesting problems in Condensed Matter Physics, in particular those relevant to high-temperature superconductors. The content of the first six chapters is roughly what I covered in that class. The remaining four chapters were developed after January 1, 1990. Thus, that material is largely the culprit for this book being one year late! During 1990 I had to constantly struggle between finalizing the book and doing research that I just could not pass on. The result is that the book is one year late and I was late on every single paper that I thought was important! Thus, I have to agree with the opinion voiced so many times by other people who made the same mistake I did and say, don’t ever write a book! Nevertheless, although the experience had its moments of satisfaction, none was like today’s when I am finally done with it.

This book exists because of the physics I learned from so many people, but it is only a pale reflection of what I learned from them. I must thank my colleague Michael Stone, from whom I have learned so much. I am also indebted to Steven Kivelson, Fidel Schaposnik, and Xiao-Gang Wen, who not only informed me on many of the subjects which are discussed here but, also, more importantly, did not get too angry with me for not writing the papers I still owe them.

In the last chapter we introduced the concept of valence-bond states and discussed several quantum disordered phases in this language. Here we will see that the quantum fluctuations of valence-bond systems are best captured in terms of a much simpler effective theory, the quantum-dimer models. An understanding of these types of phases is best accomplished in terms of gauge theories. The phases of gauge theories and their topological properties will allow us to introduce the concept of a topological phase of matter in a precise way.

Fluctuations of valence bonds: quantum-dimer models

The valence-bond crystal of Section 8.5 has a spin-correlation length of the order of one lattice constant. It represents a quantum paramagnet. However, it is not a translationally invariant state, unlike the equal-amplitude short-range RVB state. It has crystalline order of its valence bonds and it is a four-fold degenerate state.

Classical and quantum criticality

In most cases the phases of quantum field theories, in particular those of interest in condensed matter physics, can be described in terms of the behavior of local observables, such as order parameters or currents that transform properly under the symmetries of the theory. Quantum and thermal phase transitions are characterized by the behavior of these observables as a function of temperature and of the coupling constants of the theory. The phase transitions themselves, quantum or thermal, are classified into universality classes, which are represented by the critical exponents which specify the scaling laws of the expectation values of the observables. Historically, the development of this approach to critical behavior goes back to the Landau theory of critical behavior. It acquired its most complete form with the development of the renormalization group (RG) in the late 1960s and early 1970s. It is the centerpiece of Wilson’s approach to quantum field theory, in which all local quantum field theories are defined by the scaling regime of a physical system near a continuous phase transition. From this point of view there is no fundamental difference between classical (or thermal) phase transitions, which are described by the theory of classical critical behavior, and quantum phase transitions.

For example, the expectation value of a local order parameter M as the thermal phase transition is approached from below behaves as M ~ (Tc − T)β. Here Tc is the critical temperature and β is a critical exponent that depends on the universality class of the thermal phase transition and on the dimensionality of space. While quantum mechanics can play a key role in the existence of the ordered phase, e.g. superfluidity and superconductivity are macroscopic manifestations of essentially quantum-mechanical phenomena, the thermal transition itself is governed entirely by classical statistical mechanics, and quantum mechanics plays a role in setting the value of non-universal quantities such as the critical temperature, etc. On the other hand, in the case of a quantum phase transition, the order parameter M has a similar scaling behavior as a function of the coupling constant, M~(gc−g)β˜, where g is the coupling constant, gc is the critical coupling constant, and β˜ is a critical exponent that depends on the universality class of the quantum phase transition. Here we assume that M has a non-vanishing expectation value only for g < gc.

Anyon superconductivity

In this chapter we will consider the problemof predicting the behavior of an assembly of particlesobeying fractional statistics. We have already considered the problem of the quantum mechanics of systems of anyons. However, we did not consider what new phenomena may arise if the system has a macroscopic number of anyons present. At the time of writing, the physical reality of this problem is still unclear. However, this is such a fascinating problem that we will discuss it despite the lack of firm experimental support for the model.

There are two different physical situations in which the problem of anyons at finite density is important. Halperin, (1984) observed that the quasiparticles of the Laughlin state for the FQHE obeyed fractional statistics (i.e. they are anyons). In Chapter 13 we will discuss Halperin’s theory. Furthermore, Halperin and Haldane suggested that, for filling fractions of a Landau level different from the 1/m Laughlin sequence, the ground state of a 2D electron gas in a strong magnetic field could be understood as a Laughlin state of anyons. Shortly afterwards, Arovas, Schrieffer, Wilczek, and Zee (Arovas et al., 1985) studied the high-temperature behavior of a gas of anyons and calculated the second virial coefficient.