A brief introduction to Fourier series and transforms, with explicit expressions for a selected number of functions.

A concise, but fairly comprehensive summary of rotation operators, angular momentum and Wigner functions, with their combination expressions and an explicit tabulation up to order four.

A concise introduction to the scattering of X-rays from the distribution of electrons in a certain molecular system, showing the relation between measurable scattered intensities and structures described at atomistic level.

This chapter presents a full microscopic description of ordering in anisotropic fluids. A systematic introduction to the order parameters for uniaxial and biaxial nematics and smectics is provided, with attention to their physical significance as well as their determination from experiments (e.g. Linear Dichroism, Fluorescence Depolarization, NMR), with explicit examples, and from computer simulations.

This chapter discusses the pair correlations appropriate for various type of liquid crystals employing rotational invariants. A link with experimental techniques and examples from computer simulations are provided.

Chapter 6 studies the Josephson tunneling effect in a superconductor-insulator-superconductor junction. The d-wave energy gap gives rise to a geometry dependent phase factor in the tunneling current. This leads to a unique phase-sensitive tool for experimentally detecting the d-wave pairing symmetry through a corner-sharing or tri-crystal junction. It is this kind of measurement that yields the strongest evidence for identifying the pairing symmetry in cuprate superconductors. The paramagnetic Meissner effect is discussed at the end of the chapter.

Chapter 3 derives the gap equation and determines the critical transition temperature as well as the zero-temperature energy gap as a function of coupling constant for d-wave superconductors. The energy dependence of the density of states and its effects on the temperature dependence of the gap function, entropy and other thermodynamic quantities are also discussed. Low energy nodal excitations lead to characteristic power-law behaviors in the specific heat or other thermodynamic response functions of d-wave superconductors at low temperatures, in contrast to the activated behaviors in s-wave superconductors. The probability density current and charge density current operators of d-wave quasiparticles, together with the gap operators in the continuum limit, are derived and discussed with the BdG framework.

Starting from a brief introduction to the Meissner effect and other defining properties of superconductivity, Chapter 1 recapitulates the phenomenological theories, including the two-fluid model and the Ginzburg-Landau theory, and the groundbreaking microscopic theory of Bardeen-Cooper-Schrieffer for describing this macroscopic quantum phenomenon. The Cooper pairing and other basic concepts of superconductivity, such as the gap function, off-diagonal long-range order, quasiparticle excitations, coherence length, penetration depth, type-I and type-II superconductors, and phase fluctuations are also introduced, followed by a summary on the classification and experimental identification for the pairing symmetry of high-Tc superconductors.

Chapter 14 introduces the theory of d-wave superconductors in the mixed state. It starts with a detailed derivation for the Caroli-de Gennes-Matricon vortex core states and then discusses the properties of low-lying excitations under the semi-classical approximation. The universal scaling laws for several different thermodynamic quantities are derived and compared with experimental observations for high-Tc cuprates.

Chapter 13 studies the dynamic spin response function measured by neutron scattering experiments. In particular, the magnetic resonance states revealed by the neutron scattering measurements for high-Tc cuprates in the superconducting state are discussed. It is argued that this spin resonance mode may arise either from a spin exciton excitation induced by an attractive residual spin interaction in the particle-hole channel or from a collective ?-resonance mode in the particle-particle channel which emerges in the neutron scattering spectrum thanks to the particle-hole mixing in the superconducting state.

Chapter 12 studies the property of magnetic response functions of electrons probed by nuclear magnetic resonance (NMR) experiments. The Knight shift is shown to be proportional to the real part of the local magnetic susceptibility. The spin-lattice relaxation, on the other hand, provides an effective measure of the imaginary part of the susceptibility averaged by the interaction form factor over the whole Brillouin zone. The effect of impurity scattering, particularly the impurity induced resonance states, on the NMR spectra is discussed and compared with experimental results.