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Quantum Theory of Solids Hardcover – 1964
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That said, this is a great, concise reference for anyone creating models and simulations because it bridges instructive texts and books on pure mathematical methods.
After a brief introduction to the mathematics needed in the book, the author begins in chapter 2 with a treatment of acoustic phonons, which arise from the canonical quantization of the transverse motion of a continuous elastic line under tension. This object is handled using the Lagrangian formalism, and after finding the Hamiltonian density, employing a canonical transformation, the (bosonic) creation and annihilation operators are found: phonon excitations. Both longitudinal and transverse modes are shown to exist in general. Bogoliubov transformations are then used to show how phonons may arise in a system of weakly interacting particles. The author then derives the expression for the velocity of "second sound" in a phonon gas. Experimental evidence for second sound in liquid helium was known at the time of publication, but since then evidence has accumulated in Bose gases and in certain types of crystals, such as KTaO and SrTiO. The phenomenon of second sound has also been of considerable interest in the study of nonlinear optical phenomena in smectic liquid crystals. The author also discusses the occurence of van Hove singularities in the phonon frequency distribution function, and points to their connection with Morse theory.
In chapter 3 the author concentrates his attention on plasmons, which arises from longitudinal excitations in an electron gas, and optical phonons in ionic crystals. He then extends the latter analysis to include the interaction of optical phonons with photons, which he also treats using quantum field theory, giving what he calls a quantum theory of a classical dielectric.
The theory of spin waves, or "magnons" is discussed in chapter 4, wherein the author first treats ferromagnetic magnons via the consideration of the Hamiltonian consisting of nearest-neighbor exchange and Zeeman contributions. The dispersion relation for both optical and acoustical magnons in a spin system forming a Bravais lattice is derived and compared with experiment for magnetite. The author then treats antiferromagnetic magnons and discusses the zero-point sublattice magnetization and the heat capacity of antiferromagnets. He then returns to ferromagnetic magnons but from a more macroscopic point of view, treating the magnetization as a macroscopic field, rather than dealing with individual spins. Lastly, he considers the excitation of ferromagnetic magnons by parallel pumping and the temperature dependence of effective exchange.
After a short review of the Hartree-Fock approximation in chapter 5, the author considers the all-important electron gas in chapter 6. The electron gas, particularly in two dimensions, has been the subject of great interest since this book was first published, not only because of its technological importance, but also its role in the quantum Hall effect and the fractional quantum Hall effect. Although density functional and renormalization group methods are the current favored ones for studying the electron gas, readers can still gain much from the reading of the chapter. The author concentrates his attention on the approximate calculation of the correlation energy of the degenerate electron gas, particularly at high density. To do this he uses the self-consistent field approach and he exploits the frequency and wavevector dielectric constant as a tool for studying many-body interactions. Several bread-and-butter topics in quantum many-body theory appear in this chapter, such as the linked cluster expansion, which appear in other more complicated (relativistic) contexts, such as high energy physics.
The author introduces polarons in chapter 7 as a consequence of any deformation of the ideal periodic lattice of positive ion cores on the motion of conduction electrons, and notes that even the zero-point motion of phonons effects this motion. The interaction of an electron with the lattice results in a "lattice polarization field" around the electron, and the resulting composite particle is the polaron, which, as expected, has a larger effective mass then the electron in an unperturbed lattice. The electron-phonon interaction results in resistivity, results in attenuation of ultrasonic waves in metals, and results in some cases to an attractive interaction between electrons, this being one of the precursors of superconductivity. The problem of electron-phonon interaction in metals has been the subject of much study in attempts to give quantum field theory a rigorous mathematical foundation, particularly via the study of the "jellium model".
Chapter 8 is very important, and its content reveals again the age of the book. The phenomenon of superconductivity, and its description by the Bardeen-Cooper-Schrieffer theory, is known as one of the triumphs of the quantum theory of solids. Of course, when this book was published, superconducting materials at high temperature, were not known. The author though gives a detailed overview of the BCS theory, starting with the Hamiltonian for the electrons, phonons, and their first-order interactions (the strength measured by a certain real constant D). Using a canonical transformation, the author reduces the Hamiltonian to one with no off-diagonal terms of order D. This results in an expression for an electron-electron interaction which can be attractive for excitation energies in a certain range (involving the Debye energy). Keeping only this interaction in the Hamiltonian, for wave vectors that satisfy this range constraint, the author studies the properties of bound electron pairs, and shows how they bring about superconductivity. He also outlines an alternative solution to the BCS equation, using what he calls the equation-of-motion method. More modern treatments of superconductivity employ the use of Higgs fields and the renormalization group, these approaches shedding light on whether one can indeed view superconductivity as a "macroscopic manifestation of quantum physics".