The main focus of this book is many-particle systems such as electrons in a crystal. Such systems are studied within the framework of quantum mechanics, with which the reader is assumed to be familiar. Nevertheless, a brief review of this subject will provide an opportunity to establish notation and collect results that will be used later on.

The postulates

Quantummechanics is based on five postulates, listed below with some explanatory comments.

The quantum state

The quantum state of a particle, at time t, is described by a continuous, singlevalued, square-integrable wave function Ψ(r, t), where r is the position of the particle. In Dirac notation, the state is represented by a state vector, or ket, ∣Ψ(t)⟩, which is an element of a vector space V. We define a dual vector space V* whose elements, called bras, are in one-to-one correspondence with the elements of V: ket ∣α⟩ ∈ V ⟷ bra ⟨α∣ ∈ V*, as illustrated in Figure 1.1. The bra corresponding to ket c∣α⟩ is c* ⟨∣, where c*. is the complex conjugate of c. The inner product of kets ∣α⟩ and ∣β⟩ is denoted by ⟨β∣α⟩, and it is a complex number (c-number). Note that the inner product is obtained by combining a bra and a ket. By definition, ⟨β∣α⟩ = ⟨α∣β⟩*.

In this chapter we turn to phonons, photons, and their interactions with electrons. These interactions play an important role in condensed matter physics. At room temperature, the resistivity of metals results mainly from electron–phonon interaction. At low temperature, this interaction is responsible for the superconducting properties of many metals. On the other hand, the electron–photon interaction plays a dominant role in light scattering by solids, from which we derive a great deal of information about excitation modes in solids. Much of our knowledge about energy bands in crystals has been obtained through optical absorption experiments, whose interpretation relies on an understanding of how electrons and photons interact.

We begin by discussing lattice vibrations in crystals and show that, upon quantization, the vibrational modes are described in terms of phonons, which are particle-like excitations that carry energy and momentum. We will see that the effect of lattice vibrations on electronic states is to cause scattering, whereby electrons change their states by emitting or absorbing phonons. Similarly, the interaction of electrons with an electromagnetic field will be represented as scattering processes in which electrons emit or absorb photons.

A metallic crystal has a large number of mobile electrons, of the order of Avogadro's number, and a correspondingly large number of ions. If our interest is in the bulk properties of a crystal, we may take the volume V of the crystal to be infinite, and the number of electrons N to be infinite, while keeping N/V, the number density of electrons, finite; this is called the thermodynamic limit. The ions incessantly vibrate about their equilibrium positions, but due to their large mass, they move very slowly in comparison with the electrons, so that the electrons quickly adjust their state to reflect whatever positions the ions occupy at any given time. Consequently, to a good approximation, one may solve the Schrödinger equation for electrons by assuming that the ions are fixed; this is the Born–Oppenheimer approximation. The influence of the ionic vibrations on the electronic states, described through the electron–phonon interaction, may be treated by perturbation theory; this is discussed in Chapter 11.

A more drastic approximation in the description of a metal is to replace the mesh of positive ions with a uniform positive background, which results in the so-called jellium model. In a model such as this, any results obtained are necessarily qualitative in nature. In this chapter, we study the jellium model. One of our goals in this study is to show that the divergent term in the Coulomb interaction, corresponding to q = 0 (see Eq. [3.29]), is cancelled by contributions to the total energy from the positive background.

In both theory and practice, condensed matter physics is concerned with the physical properties of materials that are comprised of complex many-particle systems. Modeling the systems' behavior is essential to achieving a better understanding of the properties of these systems and their practical use in technology and industry.

Maximal knowledge about a many-particle system is gained by solving the Schrödingere quation. However, an exact solution of the Schrödinger equation is not possible, so resort is made to approximation schemes based on perturbation theory. It is generally true that, in order to properly describe the properties of an interacting many-particle system, perturbation theory must be carried out to infinite order. The best approach we have for doing so involves the use of Green's function and Feynman diagrams. Furthermore, much of our knowledge about a given complex system is obtained by measuring its response to an external probe, such as an electromagnetic field, a beam of electrons, or some other form of perturbation; its response to this perturbation is best described in terms of Green's function.

Two years ago, I set out to put together a guide that would allow advanced undergraduate and beginning graduate students in physics and electrical engineering to understand how Green's functions and Feynman diagrams are used to more accurately model complicated interactions in condensed matter physics.

Polarization

Polar crystals are generally semiconductors or insulators that, at low temperatures, have fully occupied valence bands and empty conduction bands. It is possible, however, to introduce electrons into the conduction bands. For example, absorption of photons of appropriate energy leads to the promotion of electrons from the occupied valence bands to the empty conduction bands. Raising the temperature produces a similar effect. In semiconductors, doping introduces free electrons into the lowest conduction band (or free holes into the top valence band). The electron–phonon interaction in these systems is not adequately described by the rigid-ion approximation. In an optical mode, the ions in the unit cell move relative to each other, resulting in an oscillating dipole moment which, in turn, gives rise to an electric field that acts on the electrons. The electron–LO phonon interaction in polar crystals is mainly the result of this coupling of electrons to the induced electric field.

We consider the case of a cubic crystal with two atoms per unit cell. The ionic charges are ±e*. The volume of the crystal is V, and the number of unit cells is N. In the long wavelength limit (q → 0), the two ions in the unit cell vibrate out of phase, while the displacements in one cell are almost identical to those in a neighboring cell. We denote by u+ (u−) the displacement of the positive (negative) ion within a unit cell (see Figure D.1).

A many-particle system is intrinsically quite complex. Its energy level spectrum is almost continuous, and the eigenfunctions that correspond to those energy levels are complicated functions of the particles' coordinates. The detailed form of its energy spectrum and wave functions is neither exactly calculable nor measurable; hence, we shall not be concerned with it.

In a typical experimental measurement that involves a many-particle system, a system in equilibrium is weakly perturbed in one or more ways: a particle may be added or removed, a weak electromagnetic field may be applied, a beam of electrons or neutrons may strike the system, a thermal gradient may be established across the system, and so on. Rather than attempting to calculate the full spectrum of a many particle system, it is more useful to concentrate on understanding how a system responds to such external perturbations. The method of Green's function serves this purpose well. In this chapter, we focus on real-time functions for systems in equilibrium. Imaginary-time functions will be introduced in Chapter 8. For systems out of equilibrium, such as those featuring a metallic island between two metal electrodes and an applied bias voltage that causes current to flow through the island, another formalism, that of the nonequililbrium Green's function, is needed; it will be discussed in Chapter 13.

Historically, quantization of the motion of particles was developed first. The state was described by a wave function and observables by operators. When dealing with interactions between particles and fields, such as the electromagnetic field, the fields were treated classically. Classical field equations look like the quantum mechanical equations for the wave function of the field quanta. For example, the Klein–Gordon classical field equation is similar to the quantum mechanical wave equation for a relativistic spinless particle. Quantizing the fields, leading to quantum field theory, appears to be quantizing a theory that has already been quantized; hence the name “second quantization.” In reality, there is only one quantization and one quantum theory.

The method of second quantization is important in the study of many-particle systems. It enables us to express many-body operators in terms of creation and annihilation operators, thus rendering calculations less cumbersome. Moreover, the method makes it possible to treat systems with a variable number of particles; that is why the method initially emerged in the context of quantum field theory.

In Chapter 1 we indicated that any one-particle wave function may be expanded in a complete set of states. In this chapter, we show that products of single-particle states, when properly symmetrized, form an orthonormal basis for the expansion of the wave function of an N-particle system.

Superconductivity was discovered in 1911 by H. Kamerlingh Onnes soon after he succeeded in liquefying helium (Onnes, 1911). He observed that the resistivity of mercury dropped suddenly as its temperature was lowered below a certain critical value TC (for Hg, TC = 4.2 K). Over the years, it was found that many additional elements and compounds similarly transition to a superconducting state. In this state, materials exhibit properties that are strikingly different from the normal state. Below we discuss the most important features of superconductors.

Properties of superconductors

The first important property of a material that undergoes a superconducting transition is that its resistivity drops to zero belowa critical temperature (see Figure 12.1). In a superconducting ring, a persistent electric current flows without any observable attenuation for as long as one is willing to watch.