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.

Preface to the second edition

I am extremely happy to, at long last, be able to present the second edition of this book. In spite of what I stated in the preface of the 1991 edition, I ended up not only writing a second edition but, in a sense, a new book. So one can say, once again, that we have met the enemy and it is us. I have been pleased that the 1991 edition of this book was appreciated by many people who found it useful and stimulating. I am really happy that my effort was not in vain.

My motivation for writing this book, in 1991 and now, was to present quantum field theory as a conceptual framework to understand problems in condensed matter physics that cannot be described perturbatively, and hence do not admit a straightforward reduction to some non-interacting problem. In essence, almost all interesting problems in condensed matter physics have this character. Two prime examples of problems of this type in condensed matter physics that developed in the late 1980s, and even more so in the 1990s, are the understanding of high-temperature superconductors and the quantum Hall effects. In both areas field theory played (and plays) a central role. If anything, the use of these ideas has become widespread and increasingly plays a key role. It was lucky that the first edition of this book appeared at just about the right time, even though this meant that I had to miss out on research that was and still is important. This was probably the only time that I was on time, as people who know me can relate. Much has happened since the first edition appeared in print. The problem of the quantum Hall effects has developed into a full-fledged framework to understand topological phases of matter. Although it is still an unsolved problem, the research in high-temperature superconductors (and similar problems) has motivated theorists to look for new ways to think of these problems, and the ideas of quantum field theory have played a central role. The concepts, and subtleties, of gauge theory have come to play a key role in many areas, particularly in frustrated quantum magnetism. The interactions between condensed matter and other areas of physics, particularly high-energy physics and string theory, have become more important. Concepts in topology and other areas of mathematics rarely frequented by condensed matter physicists have also entered the field with full force. More recent developments have seen the incorporation of ideas of general relativity and quantum entanglement into the field.

Topological insulators and topological band structures

The term topological insulator refers to a novel (in 2011) class of solid-state systems that have quantized transport properties due to topological properties of their band structures. In this chapter I will provide a description of the salient ideas behind this new and rapidly growing field. I will certainly not attempt to be exhaustive in the presentation. Several specialized reviews have recently become available and the reader is referred to them for more details (including an extensive list of references) (Hasan and Kane 2010; Hasan and Moore, 2011).

What is a topological insulator? It is an electronic system that is an insulator but whose band structure is characterized by a topological invariant, i.e. a number that in general is quantized to be an integer. As such, states of this type are robust in the sense that their physical properties are stable (unchanged) under the action of local perturbations of finite size. From this definition it follows that the properties of topological insulators can be characterized at the level of free-fermion systems and are not necessarily the result of strong-correlation physics. They are a generalization of the conceptual framework behind the integer quantum Hall states. However, in spite of the topological properties of their band structures, the ground states of topological insulators are essentially unique and, even when degeneracies may be present, they do not depend on the topology of the space. Hence topological insulators are not topological fluids in the sense of the fractional quantum Hall states or of the deconfined gauge theories (and spin liquids). It is quite likely that, at least in two dimensions, there may exist topological fluids that are generalizations of topological insulators. At the time of writing this is an open area of research whose future is difficult to predict.

In this chapter we will develop the effective-field theory of topological fluids, focusing on the fractional quantum Hall states as the prototype.

Quantum Hall fluids on a torus

Quantum Hall states are topological electron fluids whose properties depend on the topology of the surface on which the electrons are bound. In this section we will discuss the case of a fluid confined to a 2D torus. Although considering a 2D electron gas on a torus is of little experimental value, it is a great conceptual (and numerical) tool.

For the Laughlin states we have discussed there is the implicit assumption that the electrons are confined in a simply connected region of the plane by an external confining potential. Thus, the surface on which the electrons live has the topology of a disk or, which is equivalent, a sphere. In this geometry the ground state is unique.

In Section 12.7 we discussed the state of a free-fermion system on a torus with filling factor ν = 1 (a full Landau level), and we showed that the ground state on a torus is also unique. The reason for this is that the filling factor is 1. Thus, in a translationally invariant system the motion of the center of mass of the electron fluid decouples from the relative motion of the electrons in the fluid. The motion of the center of mass is that of a single particle moving in a magnetic field with one flux quantum. Thus the state is unique.

Generalized spin chains: the Haldane conjecture

The phenomenology which emerges from the spin one-half Heisenberg antiferromagnetic chain is quite striking: there is no long-range order, and there are gapless states, in particular, gapless spinless fermions (which, in the Heisenberg picture, are solitons). From the point of view of the Hubbard model, the Heisenberg model occurs at infinite coupling, where the charge-bearing degrees of freedom acquire a gap that is infinitely large. Thus spin and charge degrees of freedom are separated and the spin sector is at a critical point. This phenomenology inspired Anderson, (1987) to propose a similar picture for the two-dimensional systems, the resonating-valence-bond (RVB) picture. However, most of this picture surely should not generalize. Critical points are not generic and, in general, it is not possible to have gapless states without the spontaneous breaking of a continuous symmetry except in one dimension due to the Mermin–Wagner theorem. In higher dimensions gapless states without a broken symmetry may be possible in a Coulomb phase of a gauge theory with a continuous gauge group. Thus, the 1D spin one-half case may be more the exception than the rule. For instance, it may be possible that the system is in a state without long-range order, which is likely to be massive. For this reason, it is important to consider generalizations of the Heisenberg model. This problem has been studied extensively. Two different approaches have been considered in one dimension: (a) enlarging the representation (higher spin, same symmetry group SU(2)) and (b) higher symmetry groups (SU(N), for instance).

In an incompressible quantum fluid, such as the Laughlin state, the fluctuations in the bulk induce fluctuations at the boundary. While the local fluctuations in the bulk are associated with local changes in the density, the fluctuations of the states at the boundary are associated with changes in the shape of the “droplet” of the electron fluid, shown in Fig. 15.1. These “edge waves” are the only gapless excitations of the system. It may seem surprising that an incompressible fluid may have gapless modes at the surface, although this is quite common in conventional fluids such as water! In the FQHE the gaplessness arises from the fact that the geometric edge of the fluid coincides with the locus of points in which the Fermi energy crosses the external potential which confines the fluid. Thus, the boundary of the fluid behaves like a “Fermi surface” and, as we move from the edge and into the bulk, we get deeper and deeper in the Fermi sea of occupied states. Because of the presence of the magnetic field, the edge waves are chiral excitations that move at the drift velocity of the particles at that point. Thus, edge states move only in one direction, which is specified by the magnetic field. The importance of the edge states to the observability of the quantum Hall effect was first emphasized by Halperin (1982). The description of the chiral quantum dynamics of the edge states is due to X. G. Wen (Wen, 1990a, 1990b, 1991b) and M. Stone (Stone, 1991).

Edge states of integer quantum Hall fluids

Let us consider the physics of the edge waves in the context of the simplest system: non-interacting electrons filling up the lowest Landau level (ν = 1). Strictly speaking, we are discussing the behavior of the edge states in a system with an integer quantum Hall effect (IQHE). However, at least within a mean (or average)-field approximation, the fractional quantum Hall effect can also be regarded as an IQHE of an equivalent system of fermions. We will discuss this point of view (originally due to Jain) in this chapter, where we present the Chern–Simons approach to the FQHE. In this section we follow the methods of Stone (1991).

In this chapter we discuss the theory of the fractional quantum Hall effect (FQHE). The explanation of this phenomenon has required the development of completely new ideas and methods. The concept of fractional statistics has become a crucial element of the theory.

The physical system involves fermions in strong correlation in the absence of time-reversal symmetry. The treatment of systems with these features cannot be achieved successfully within the conventional Hartree–Fock approach to correlations in condensed matter physics. A new condensed state of matter, the Laughlin state, had to be discovered.

The Chern–Simons gauge theory, which has already been discussed in Chapter 10, has come to play an essential role in the theory of the FQHE, both as a way to describe the low-energy phenomena and as a theoretical tool to explain the most important features of the problem.

We begin with a detailed description of the theory of the Laughlin wave function, which is followed by the field-theory approaches to the FQHE.

The Laughlin wave function

In the last two sections of the previous chapter we considered the problem of electrons moving on a 2D surface in the presence of a perpendicular magnetic field. We assumed that the electron density was such that an integer number of Landau levels (or bands) would be completely filled. Because the system has an energy gap, the interactions do not play a very important role. In fact, a perturbative expansion (in powers of the coupling constant) around a state with one filled Landau level (or more) is likely to be well behaved. Since all processes involve exciting one or several electrons across the gap, the energy denominators are always different from zero. The ground-state wave function for the interacting system is smoothly connected to the ground-state wave function of the non-interacting system. The arguments of the last three sections of Chapter 12 indicate that the topological properties of the wave function for the interacting and the non-interacting systems will then be the same. In other words, naive perturbation theory is a good approximation in this case.

The Hubbard model was originally introduced as the simplest system which may exhibit an insulating (Mott) state. This state is the result of strong electron–electron interactions. In this chapter we consider the Hubbard model at half-filling. The main goal here is the study of the magnetic properties of its ground state. Apart from an exact solution in one dimension, no exact results are available for this problem. This leads to the use of several approximations. The most popular one, and the oldest, is the mean-field theory (MFT). In the MFT one has the bias that the ground state does have some sort of magnetic order (i.e. ferromagnetic, Néel antiferromagnetic, etc.). The problem is then usually solved by means of a variational ansatz. However, one is usually interested in more than just the ground-state energy, which, after all, is not directly measurable and depends very sensitively on the properties at short distances. Most often we wish to evaluate the long-distance, low-frequency, properties of the correlation and response functions of this theory. Moreover, in some cases, such as in one dimension, the fluctuations overwhelm the MFT predictions.

In this chapter we will consider the standard MFT (i.e. Hartree–Fock), which is expected to become accurate at weak coupling. We will consider both ferromagnetic and antiferromagnetic states. We will also rederive these results using path integrals. As a byproduct, we will also have a theory of the fluctuations: the non-linear sigma model.