The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

This chapter discusses light–matter interactions from a semiclassical point of view. By expanding the electromagnetic field into a Taylor series we derive the multipolar interaction potential and particle-field Hamiltonian. Then, using the Green function formalism, we calculate the fields of an oscillating dipole and, based on Poynting’s theorem, derive a general expression for the rate of energy dissipation in an arbitrary environment. This expression leads to the concept of local density of states (LDOS) and provides a direct link to spontaneous emission and atomic decay rates. The rate of energy dissipation of an oscillating dipole is also used to derive the absorption cross-section in terms of the polarizability. By accounting for radiation reaction and scattering losses, we obtain a compact expression for the dynamic polarizability. Dipole radiation and atomic decay rates can be enhanced via LDOS engineering, and the enhancement factor is referred to as the Purcell factor. We show that if the LDOS gets enhanced in a certain frequency range, it must be reduced in other frequency ranges, a feature described by the LDOS sum rule. After discussing the properties of a single dipole, we continue with analyzing the interaction between multiple dipoles. We derive the interaction potential and calculate the energy transfer rate between dipoles. For short distances we recover the famous Förster energy transfer formula. If the interaction energy becomes sufficiently large, we enter the regime of strong coupling, which gives rise to hybridized and delocalized modes, level splittings, and entanglement.

The chapter introduces the field of plasmonics, focusing on the unique optical properties of noble metals. We begin by describing noble metals as plasmas, introducing concepts such as electronic screening and the ponderomotive force. We then discuss the local optical response of noble metals through their frequency-dependent dielectric function. The chapter progresses to surface plasmons at plane interfaces, detailing the properties of surface plasmon polaritons (SPPs), methods for exciting SPPs, and their practical applications in sensors, particularly their sensitivity and utility in biochemical sensing. Next, we focus on plasmons supported by nanowires and nanoparticles, utilizing the quasistatic approximation and standing plasmon waves. We analyze plasmon resonances in complex nanostructures by introducing the concept of plasmon hybridization. These resonances play a crucial role in surface-enhanced Raman scattering (SERS), enabling the detection of low-concentration analytes. We further explore nonlinear plasmonics, which leads to phenomena such as harmonic generation and four-wave mixing. Finally, we address nonlocal plasmonics, discussing the impact of spatial dispersion and mesoscopic boundary conditions on plasmonic responses at very small scales, highlighting the quantum effects at the metal surface.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

This chapter reviews the main concepts of electromagnetic theory relevant for the understanding of this textbook. Based on Maxwell’s equations, we derive the wave equation and discuss homogeneous solutions, such as plane waves and evanescent waves. We derive the boundary conditions at interfaces between homogeneous media and the Fresnel reflection and transmission coefficients. We discuss energy conservation, causality, and reciprocity of electromagnetic fields. Point response functions are introduced (Green functions) in order to derive the inhomogeneous solution of the wave equation. The chapter concludes with the angular spectrum representation, a framework that allows arbitrary fields to be described as a superposition of plane and evanescent waves.

In practice, a radiating source is most commonly placed close to a surface or multiple interfaces. Examples are antennas mounted over ground or molecules placed on dielectric surfaces or waveguides. The topic has a long history dating back to Arnold Sommerfeld’s paper in 1909. To derive the fields of an arbitrary oriented dipole over a layered medium we have to find the corresponding Green function. We start by decomposing the free-space dyadic Green function into $s$ and $p$ polarized parts and then evaluate reflection and transmission for the two polarizations separately. Once the Green function of the layered reference system is found, we proceed to derive the radiated power and the far-fields of the dipole. We analyze the radiation patterns and the modes into which the energy is most effectively coupled. For dipoles over dielectric half-spaces, we find that evanescent field components couple predominantly into supercritical angles, giving rise to what is termed “forbidden light.” We discuss how recorded radiation patterns can be used to determine the orientation of the radiating dipole, such as the orientation of molecules fluorescing near a dielectric surface. The chapter concludes by reviewing the image dipole method and discussing its validity.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The properties of optical materials are determined by the fundamental constituents of matter. In this chapter we discuss how to design materials with unusual optical properties by arranging meta-atoms, smaller than the wavelength of light, fabricated using nanotechnology. Meta-atoms arranged in periodic arrays with lattice constants of the order of the wavelength lead to photonic crystals. Photons in photonic crystals behave analogous to electrons in regular crystals, allowing principles from solid-state physics, such as doping, to be carried over. This results in fascinating effects such as photonic bandgaps and localized states of light. Metamaterials arise if meta-atoms are densely packed such that light propagates as if it were in a homogeneous medium. By tuning the properties of the meta-atoms, these materials can be tailored to exhibit exotic optical properties such as negative or near-zero refractive index. Finally, we introduce metasurfaces which directly mold the flow of light – a property that can be used to create ultraflat optical elements.

This introductory chapter sets the stage for the research field of nano-optics. It introduces the fundamental concept of localizing light beyond the diffraction limit through the superposition of propagating and evanescent waves, emphasizing the critical role of evanescent waves. Additionally, it provides a historical overview of the key developments that have shaped nano-optics, and outlines the scope of the book.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.