Electron in a strong magnetic field

General remarks

Some aspects of electron motion in a two-dimensional (2D) disordered metal have been considered in Chapter 5. These were effects that could be studied by using the perturbation theory in diffusion modes. The renormalization group scheme is a way to sum up a certain class of graphs and is a straightforward extension of the simple perturbation theory. Although the first quantum correction, Eq. (5.23), contains many interesting effects that can be and have been confirmed by numerous experiments, some other interesting effects cannot be obtained in this simple manner.

One of the most interesting phenomena occurring in two dimensions is the quantum Hall effect (von Klitzing, Dorda, and Pepper (1980)). Since its discovery a lot of theoretical and experimental activity has been devoted to studying this effect and related properties of 2D electron gases. It makes no sense to review this direction of research here because many interesting and comprehensive reviews and books already exist (see, e.g., Prange and Girvin (1990), Büttiker (1992), Stone (1992), Aoki (1986), Janßen et al. (1994)). The aim of this section is to demonstrate only how the supersymmetry technique can help in studying electron motion in a 2D disordered metal.

In fact, in discussing theoretical aspects of the quantum Hall effect one should distinguish between the integer quantum Hall effect (occurring at integer filling factor vf) and the fractional effect discovered by Tsui, Stormer, and Gossard (1982).

One-dimensional σ-model

One-dimensional σ-model for disorder problems

Most of the results of Chapters 6–10 have been obtained by using the zero-dimensional (0D) σ-model and we have seen that many rather different physical problems can be treated within this approximation successfully. Nevertheless, for some problems that can be described by the supermatrix σ-model the 0D version is not applicable. For example, localization in two-dimensional disordered metals was discussed in Chapter 5 and it had to be described by the two-dimensional (2D) σ-model. The problem is rather complicated and the solution presented there is not complete.

There are many problems of disordered metals and quantum chaos that can be reduced to a study of the one-dimensional (ID) σ-model. In contrast to those with higher dimensions, the ID σ-model can be studied exactly by the transfer matrix technique, and very often it is possible to get explicit final results although the calculations are somewhat more difficult than in the 0D case. By now, the procedure of computation of different correlation functions with the ID σ-model is well worked out, but before presenting it in detail let us discuss physical problems that can be treated in this way.

It is natural to suppose that the ID σ-model describes one-dimensional disordered metals. However, when using the term one-dimensional one should distinguish between truly one-dimensional chains and microscopically three-dimensional metals with a one-dimensional geometry of the sample.

What is supermathematics?

All supersymmetric theories are based on the use of anticommuting classical variables first introduced by Grassmann in the last century. At first glance, these objects look very artificial and seem to have no relation to the real world. There is a certain threshold for physicists to start using the Grassmann anticommuting variables for calculations because one expects the game to have very unusual rules. Surprisingly, it is not true, and provided proper definitions are given, one can simply generalize conventional mathematical constructions so that it is possible to treat both commuting and anticommuting variables on an equal footing. Sometimes the corresponding branch of mathematics is called supermathematics.

Of course, the main purpose of this book is to consider different physical results obtained with the use of the Grassmann variables, and therefore one could try to demonstrate how these variables work while making some concrete calculations. However, it seems to be more reasonable to present the basic formulae of supermathematics in one place, first, because it may be the best way to get used to the anticommuting variables, and, second, because one can see that practically all the rules of operating with “superobjects” are quite standard.

Today the mathematical analysis and algebra of functions of both commuting and anticommuting variables are very well developed.

Historical remarks

The last 15–20 years has witnessed spectacular progress in the study of disordered metals and semiconductors. These systems are interesting not only from the point of view of different technical applications but also because they reveal new unusual physical properties that are very different from what one would expect in clean regular materials. Although very often thermodynamic characteristics are already quite influenced by disorder, the most remarkable effects are observed in kinetics. Of course, in many cases one may use the classical transport theory based on the Boltzmann equation for a description of electron motion. However, if the disorder is strong or temperature is low, quantum effects become important, and to construct a theory in this situation one has to start from the Schrödinger equation in a potential that is assumed to be random.

To get information about physical properties of the system one has to solve the Schrödinger equation for an arbitrary potential, calculate a physical quantity, and, at the end, average over the random potential. Sometimes it is important to have information not only about the average but also about fluctuations. In this case one has to calculate moments of the physical quantities and even an entire distribution function. Needless to say, generally speaking, this program cannot be carried out exactly even in the absence of electron–electron interaction and one should use different approximation schemes.

Random matrix theory

General formulation

According to the basic principles of quantum mechanics the energy spectrum of a particle in a limited volume is discrete. The positions of the energy levels and the spacings between them depend on the boundary conditions and interactions in the system. In the simplest cases these quantities can be calculated exactly or approximately. However, often the interactions are so complicated that calculations for the levels become impossible. On the other hand, the complexity and variety of interactions lead to the idea of a statistical description in which information about separate levels is neglected and only averaged quantities are studied. Density of states, energy level and wave function correlations, and the like, can be so considered. The analogous approach is used in statistical physics, where information about separate particles is neglected and only averages over large number of particles are calculated.

The idea of statistical description of the energy levels was first proposed by Wigner (1951, 1958) for study of highly excited nuclear levels in complex nuclei. In such nuclei a large number of particles interact in an unknown way and it is plausible to assume that all the interactions are equally probable. Of course, the first question one can ask is what the characterization “equally probable” means and therefore one should introduce a measure for averaging.

Recent progress in physics of disordered metals and semiconductors has led to the development of theoretical methods adequate for their description. Now, it is completely clear that such disciplines of theoretical physics as theories of disorder and quantum chaos are necessary to describe, for example, modern mesoscopic quantum devices. Moreover, these disciplines are converging toward each other, an exciting theoretical development. Although a lot of information can be obtained from numerical simulations, an analytical approach unifying disorder and chaos is definitely desirable. Besides, numerical simulation is often not conclusive and one has to have an analytical tool for calculations.

Currently the most efficient analytical method enabling us to achieve both goals is the supersymmetry technique, and many problems of disorder and chaos can be studied with a supermatrix nonlinear σ-model. The number of publications using the supersymmetry technique has been growing fast in the last 2–3 years. At the same time, many people still have a hesitation to start study of the method. The main reason is that they are afraid that manipulating the Grassmann anticommuting variables is something very difficult and, what is more important, that having spent a considerable time learning the technique, they would be able only to reproduce results that could be obtained by other more standard techniques. Such an attitude is to a great extent due to absence of a self-contained literature on the subject.

Description of phase transitions

Phenomenological approach

Some basic information concerning theory of the Anderson metal–insulator transition has been given in Chapters 3 and 5. The agreement of the one-parameter scaling hypothesis of Abrahams et al. (1979) with the results of the renormalization group treatment of the nonlinear replica and supermatrix σ-models in 2 + ∈ dimensions was considered by many researchers final proof that the transition was a conventional second-order transition. The only thing that remained to be done was to compute critical exponents, and that could be done by making an expansion in ∈ and putting ∈ = 1 at the end. Other approximate schemes (Götze (1981, 1985), Vollhardt and Wölfle (1980, 1992)) lead to similar results. Although agreement between the exponents computed analytically and those extracted from numerical simulations or experiments was not always good, the validity of the one-parameter scaling description was not usually questioned.

Of course, on the basis of what is known about mesoscopic systems one cannot speak of the average conductance of a finite system, and, possibly, the entire distribution function of the conductances should be scaled. However, renormalization group treatment of the σ-model in 2 + ∈ dimensions does not lead to such a scenario. If one accepts that this approach is appropriate for studying the Anderson transition, the conclusion that the transition is a conventional second-order phase transition is inevitable.

It is clear from the material presented here that the supersymmetry technique, in particular, the supermatrix nonlinear σ-model, is an extremely efficient way of studying various problems. A natural question may be asked: Why does all this work? What is the physical meaning of the invariance of the σ-model under rotations of supermatrices Q?

These questions are not easy to answer. I cannot explain why the supermatrix σ-model enables us to get some nontrivial results that cannot be obtained by other methods currently available. The supersymmetry formalism was derived from the Schrödinger equation with a random potential. All symmetries of the σ-model appeared in the process of the derivation, but the initial Schrödinger equation does not contain them. My attitude to these questions is that all nice symmetry features of the σ-model are formal and it is difficult to attribute to them a clear physical sense.

I want to emphasize that the Grassmann anticommuting variables χi were introduced in a completely formal way. The initial Schrödinger equation did not contain them and they were introduced with the hope that they would help in the calculations. I cannot explain why the variables χi that are completely formal mathematical objects helped to get the results. However, it is not unusual for abstract mathematical objects to be useful for explicit computations.

Basic properties

The recent observations of persistent currents in small metallic rings by Lévy et al. (1990), Chandrasekhar et al. (1991), and Mailly, Chapelier, and Benoit (1993) have opened a new field of research. Although small mesoscopic systems have been under intensive study for quite a long time, they have usually been studied by making contacts with metallic leads. As a result, one could obtain a finite conductance of the system that corresponded to a finite current in the presence of a finite voltage only. The experiments by Lévy et al. (1990) and by Chandrasekhar et al. (1991) were carried out in such a way that the metallic rings remained isolated. A slowly varying magnetic field was applied and a magnetic response was measured. In this situation it became possible to observe persistent currents in quite dirty samples in which the elastic mean free path l was much shorter than the circumference L. The existence of the persistent current is possible in isolated rings only, and Mailly, Chapelier, and Benoit (1993) demonstrated how the value of the persistent current decreases when increasing a weak coupling to leads.

The intriguing question of persistent currents in metal rings enclosing a magnetic flux was discussed in the 1960s by Byers and Yang (1961), F. Bloch (1965, 1968, 1970), Schick (1968), Gunter and Imry (1969).

Reduction to a regular model, mean field theory

Regular model with “interaction”

The main task of this section is to represent the physical quantities discussed in the preceding chapter in such a form that one could average over the random potential at the beginning of the calculations. One way to do so is the replica trick suggested by Edwards and Anderson (1975) for a study of spin glasses. The first works on the application of field theoretical models to disorder problems were based on this trick (Wegner (1979), Schäfer and Wegner (1980), Efetov, Larkin, and Khmelnitskii (1980)). In the works of Wegner (1979) and Schäfer and Wegner (1980) kinetic quantities were written in terms of functional integrals over conventional numbers, whereas in the work of Efetov, Larkin, and Khmelnitskii (1980) integration over the anticommuting Grassmann variables was used. Then, in both approaches nonlinear σ-models that contained n × n matrices were derived. At the end of calculations one had to take the limit n → 0. The formalism of Efetov, Larkin, and Khmelnitskii (1980) was extended later to include electron–electron interactions (Finkelstein (1983, 1984)) and strong magnetic fields (Levine, Libby, and Pruisken (1984)).

The nonlinear supermatrix σ-model derived later has many common features with the replica σ-model, and it may seem that the calculations within all these models are equivalent.