Solid state physics grew out of applications of quantum mechanics to the problem of electron conduction in solids. This seemingly simple problem defied solution because the presence of an ion at each lattice site seemed to present an obvious impediment to conduction. How the electrons avoid the ions was thus the basic question. Although the answer to this question is well known, it does serve to illuminate the very essence of solid state physics: there is organization in the many. Each electron adjusts its wavelength to take advantage of the periodicity of the lattice. In the absence of impurities, conduction is perfect. Hence, by understanding this simple fact that periodicity implies perfect conduction, it became clear that the experimentally observed resistivity in a metal came not from electrons running into each of the ions but rather from dirt (disorder), thermal effects mediated by dynamical motion of the ions, or electron–electron interactions. This book examines each of these effects with an eye for identifying underlying organizing principles that simplify the physics of such interactions.

Spontaneously broken symmetry

The search for organizing principles that help simplify the physics of many-body systems is at the heart of modern solid state or, more generally, condensed matter physics. One such tool is symmetry. Consider the simple case of permutation symmetry typically taught in a first class in quantum mechanics. This symmetry was introduced into quantum mechanics by W. Heisenberg in the context of the indistinguishability of identical particles.

In this chapter we focus on the phenomenon of superconductivity and the Bardeen–Cooper–Schrieffer (BCS) (BCS1957) theory behind it. Superconductivity obtains when a finite fraction of the conduction electrons in a metal condense into a quantum state characterized by a unique quantum-mechanical phase. The specific value of the quantum-mechanical phase varies from one superconductor to another. The locking in of the phase of a number of electrons on the order of Avogadro's number ensures the rigidity of the superconducting state. For example, electrons in the condensate find it impossible to move individually. Rather, the whole condensate moves from one end of the sample to the other as a single unit. Likewise, electron scattering events that tend to destroy the condensate must disrupt the phase of a macroscopic number of electrons for the superconducting state to be destroyed. Hence, phase rigidity implies collective motion as well as collective destruction of a superconducting condensate. The only other physical phenomenon that arises from a similar condensation of amacroscopic number of particles into a phase-locked state is that of Bose–Einstein condensation. There is a crucial difference between these effects, however. The particles that constitute the condensate in superconductivity are Cooper pairs, which do not obey Bose statistics. In fact, it is the Pauli principle acting on the electrons comprising a Cooper pair that prevents the complete mapping of the superconducting problem onto a simple one of Bose condensation.

A problem that has always been central to solid state physics is the insulator–metal transition. The question to answer here is why some materials conduct and others do not. Metals are characterized by a non-zero dc conductivity σ(0) at zero temperature, whereas σ(0) = 0 in an insulator. There are currently three standard models that describe a transition between these two extremes. Anderson (A1958) was first to point out that scattering from a static but random potential can disrupt metallic conduction and lead to an abrupt localization of the electronic eigenstates. Mott (M1949), on the other hand, proposed that an insulating state can obtain even in a material such as NiO which possesses a partially filled valence band. The insulating state arises from strong electron correlations which induce a gap at the Fermi energy. The closing of the gap, signaling the onset of a metallic state, results typically in the intermediate-coupling regime in which the kinetic energy effects can destroy the ordering tendencies of the potential energy. Finally, a structural transition in which the lattice periodicity doubles can also thwart metallic transport. While all of these mechanisms are of considerable interest in their own right, our focus in this chapter will be the disorder-driven insulator–metal or Anderson transition.

We start by reviewing the essential physics and some of the key controversies surrounding the disorder-induced localization transition. Two controversies we address are the role of perturbation theory and whether or not the conductivity is continuous in the vicinity of the Anderson transition.

In the previous chapter, we discussed both the insulator–superconductor and the insulator–metal transitions. As such transitions are disorder or magnetic-field tuned, thermal fluctuations play no role. Phase transitions of the insulator–superconductor or insulator–metal type are called quantum phase transitions. Such phase transitions are not controlled by changing the temperature, as in the melting of ice or the λ-point of liquid helium, but rather by changing some system parameter, such as the number of defects or the concentration of charge carriers. In all such instances, the tuning parameter transforms the system between quantum mechanical states that either look different topologically (as in the transition between localized and extended electronic states) or have distinctly different magnetic properties. As quantum mechanics underlies such phase transitions, all quantum phase transitions obtain at the absolute zero of temperature and thus are governed by a T = 0 quantum critical point. While initially surprising, this state of affairs is expected, because quantum mechanics is explicitly a zero-temperature theory of matter. Of course, this is of no surprise to chemists who have known for quite some time that numerous materials can exhibit vastly distinct properties simply by changing the chemical composition and, most importantly, that such transformations persist down to zero temperature. Common examples include turning insulators such as the layered cuprates into superconductors simply by chemical doping or semiconductors into metals once again by doping, or ferromagnets such as Li(Ho, Tb)x Y1-x F4 into a spin glass (AR1998) by altering x.

In writing the second edition of this text, I have tried to accomplish three things. First, correct all the typos in the first edition. This has turned out to be somewhat harder than I had anticipated. While I am certain my proofreaders and I corrected all mistakes we could find, that might not have been sufficient. As there will undoubtedly be a second printing, simply email me any errors you might find at dimer@illinois.edu. Second, include all the material that should have been in the first edition but that I had given up on writing. This includes Green functions, Luttinger's theorem, renormalization of short-range interactions for Fermi liquids, and symmetry. In keeping with this being a physics rather than a technique or mathematics tract, these subjects are interwoven wherever they are first needed. For example, the section on Green functions is in Chapter 7 where the Anderson impurity problem is treated. For completeness, Luttinger's theorem is also presented in the same chapter but in an appendix. Third, include new material that reflects the fast-moving pace of ħ = 1 research in condensed matter physics. Here I made a judgement based on what I anticipate students would find most useful. Since there are no texts that present the pedagogy of topological insulators (though some excellent review articles exist) and Mott insulators, I chose to focus on those topics. In writing the topological insulator section, I have tried to stick to the formulations that require the fewest definitions and new concepts since the physics of these systems is inherently simple.

Prior to 1986, research on strongly interacting electron systems was a fringe subject in solid state physics. The discovery of high-temperature superconductivity in the copperoxide ceramics changed this perception radically. The reason is simple. Undoped, the copper-oxide materials contain a partially filled d-band of electrons (A1987). Band theory tells us that partially filled bands conduct. However, the undoped cuprates are extremely good insulators. Although they become conducting after they are doped, they exhibit spectral weight redistributions over large energy scales that cannot be understood within the traditional theory of metals. It is this failure of band theory to predict insulating states in certain half-filled bands that Mott addressed in 1949 (M1949). Mott's analysis grew out of his study of NiO which contains two unpaire d electrons per unit cell but insulates nonetheless. Similar insulating states are found in most transitionmetal oxides, most notably VO2 and V2O3. Since band theory failed, Mott zeroed in on the electron correlations as the root cause of the insulating state in NiO. This line of inquiry has yielded some of the most subtle results and enigmatic problems in solid state physics. It is the physics of the Mott state and how it plays out in the cuprates and other systems that we describe in this chapter.

Any problem in solid state physics can be said to be solved once one has isolated the propagating degrees of freedom. These are the excitations that make the Lagrangian quadratic and give rise to pole-like singularities in the single-particle Green function.

Matthias and co-workers, in a series of electron spin resonance (ESR) and nuclear magnetic resonance (NMR) experiments on non-magnetic metals – metals with no permanent magnetic moment – observed surprising evidence for long-lived local spin packets in the ESR lineshape (M1960). These data indicated the persistence of local magnetic moments. The magnetic moment was quickly traced to the presence of small amounts of magnetic impurities. While various systems were studied, such as Mn, Fe, and other iron group impurities in host materials such as Cu, Ag, and Au, the common ingredient shared by all the impurity ions is that they possessed one or more vacant inner-shell orbitals. In addition, the experiments demonstrated that varying the kind and amount of the magnetic impurities did not always result in the formation of local magnetic moments in non-magnetic metals. This finding added to the intrigue and established the question of the formation of local magnetic moments as central to understanding magnetism and transport in solids. In this chapter, we describe the origin of local moments, focusing primarily on Anderson's model (A1961), the model that rose to the fore as the standard microscopic view of local magnetic moment formation in metals.

Local moments: phenomenology

An impurity in a non-magnetic metal can give rise to a local moment if an electronic state on the impurity is singly occupied, at least on the time scale of the experiment. Friedel (F1958) was the first to introduce a phenomenological model to explain the onset of local moments.

When an electron gas is confined to move at the interface between two semiconductors and a magnetic field is applied perpendicular to the plane, a new state of matter (TSG1982) arises at sufficiently low temperatures. This state of matter is unique in condensed matter physics in that it has a gap to all excitations and exhibits fractional statistics. It is generally referred to as an incompressible quantum liquid or as a Laughlin liquid (L1983), in reference to the architect of this state. While the Laughlin state is mediated by the mutual repulsions among the electrons, it is the presence of the large perpendicular magnetic field that leads to the incompressible nature of this new many-body state. The precursor to this state is the integer quantum Hall state. In this state, disorder and the magnetic field conspire to limit the relevant charge transport to a narrow strip around the rim of the sample. The novel feature of this rim or edge current is that it is quantized in integer multiples of e2/h (KDP1980). The equivalent current in the Laughlin state is still quantized but rather in fractional multiples of e2/h. We present in this chapter the phenomenology and the mathematical description needed to understand the essential physics of both of these effects.

As we will see, topology is an integral part of the quantum Hall effect. Regardless of the geometry or smooth changes in the Hamiltonian, the quantization of the conductance depends solely on the existence of edge states which have a well-defined chirality.

In the previous chapter we developed a mean-field criterion for local magnetic moment formation in a metal. As mean-field theory is valid typically at high temperatures, we anticipate that at low temperatures, significant departures from this treatment occur. The questions we focus on in this chapter are: (1) how does the presence of local magnetic moments affect the low-temperature transport and magnetic properties of the host metal, and (2) what is the fate of local magnetic moments at low temperatures in a metal? These questions are of extreme experimental importance because it has been known since the early 1930s that the resistivity of a host metal such as Cu with trace amounts of magnetic impurities, typically Fe, reaches a minimum and then increases as – ln T as the temperature subsequently decreases.