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This chapter finally deals with the concept of spin glasses. The intention is not to provide anything approaching a thorough history of the subject. The field today is broad, with threads and subthreads extending in a multitude of different directions. Rather, the chapter focuses on a relatively narrow part of the overall subject. It discusses some of the history of their discovery, their basic properties and experimental phenomenology, and some of the mysteries surrounding them. It introduces some of the basic theoretical constructs that underlie much of the discussion in later chapters. Topics covered include dilute magnetic alloys and the Kondo effect, nonequilibrium and dynamical behavior, mechanisms underlying spin glass behavior, the Edwards–Anderson Hamiltonian, frustration, dimensionality and phase transitions, broken symmetry and the Edwards–Anderson Order Parameter, and energy landscapes and metastability.

This chapter refers to the previous chapter in which the discontinuous behavior of thermodynamic functions at a phase transition was referred to as “carefully controlled conditions.” To proceed to the next part of the story, it is necessary to explain just what is meant by this. To do so, the chapter turns to the central notion of thermodynamic equilibrium. It chapter introduces systems with quenched disorder, namely, ordinary glasses. This requires an explanation of the central notions of equilibrium and nonequilibrium. The chapter shows how large a gap remains in our understanding of the condensed state, and how powerful is the challenge to conventional statistical mechanics presented by quenched disorder.

This chapter introduces mean field theory, both as a general class of models and in its specific incarnation in spin glasses, the Sherrington–Kirkpatrick model. This is undoubtedly the most theoretically studied spin glass model by far, and the best understood. For the nonphysicist the going may get a little heavy in this chapter once replica symmetry breaking is introduced, with its attendant features of many states, non-self-averaging, and ultrametricity—but an attempt is made to define and explain what all of these things mean and why replica symmetry breaking represents such a radical departure from more conventional and familiar modes of symmetry breaking. While this is a central part of the story of spin glasses proper, the nonphysicist who wants to skip the technical details can safely omit certain sections in the chapter and continue on without losing the essential thread of the discussion that follows.

This chapter considers how spin glass science fits into the larger area of complexity studies. It discusses three landmark papers in the field of complexity, by Warren Weaver, Herb Simon, and Phil Anderson, respectively, and examines how the ideas they introduced might relate to the current understanding of spin glasses. It also takes a brief look at recent developments, in particular various proposals for measures of complexity, and considers how they might illuminate some features of spin glasses. It concludes by asking whether spin glasses can still be thought of as “complex systems,” and in so doing introduces a proposal for a kind of “new complexity” as it relates to spin glasses.

The chapter explains that up to this point the discussion has centered on some basic concepts of condensed matter physics as viewed through the illustrative lenses of familiar systems: liquids, crystals, and glasses. The chapter now turns to another important class of materials: magnetic systems, which are regarded as materials possessing properties that can be altered or manipulated through the application of an external magnetic field. The chapter introduces the basics of solid state magnetism, starting with the quantum mechanical property of spin, and showing how the familiar phenomenon of ferromagnetism—as well as the less familiar but equally important ones of antiferromagnetism and paramagnetism—arises. This is a necessary prelude to understanding the idea of what a spin glass is.

This chapter introduces the basic concepts and language that will be needed later on: order, symmetry, invariance, broken symmetry, Hamiltonian, condensed matter, order parameter, ground state, and several thermodynamic terms. It also presents the necessary concepts from thermodynamics and statistical mechanics that will be needed later. It boils down the latter to its most elemental and essential ingredient: that of temperature as controlling the relative probabilities of configurations of different energies. For much of statistical mechanics, all else is commentary. This is sufficient to present an intuitive understanding of why and how matter organizes itself into different phases as temperature varies, and leads to the all-important concept of a phase transition.

This chapter discusses short-range spin glasses. It considers realistic spin glass models, in particular the Edwards–Anderson (EA) model. Both the EA and the Sherrington–Kirkpatrick (SK) models are idealizations of the complicated spatial structure of spin–spin interactions in real materials. In the EA idealization, the interactions are extremely short range, occurring only between spins that are nearest neighbors in the atomic lattice. This caricatures the actual spatial structure of laboratory spin glasses, but the EA model is nevertheless believed to distill their essential physics. In contrast, the SK model is bereft of geometric structure and behaves like an EA model in the limit of infinite dimension. Since we are really interested in finite dimensions, it is important to understand not only how the phenomena exhibited by the EA model depend on dimension d, but also how the similarities and differences between the EA and SK models depend on dimension.

This chapter explores how spin glass concepts have found use in and, in some cases, further advanced areas such as computational complexity, combinatorial optimization, neural networks, protein conformational dynamics and folding, and computer science (through the introduction of new heuristic algorithms such as simulated annealing and neural-based computation, and through new approaches to analyzing hard combinatorial optimization problems). It also introduces some “short takes” on topics that space constraints prevent covering in detail, but should be at least mentioned: prebiotic evolution, Kauffman's NK model, and the maturation of the immune response. The chapter summarizes the heart of what most people mean when they refer to spin glasses as relevant to complexity. It focuses on the early, classic papers in each subject, giving the reader a flavor of each.

Spin glasses are disordered magnetic systems that have led to the development of mathematical tools with an array of real-world applications, from airline scheduling to neural networks. This book offers the most concise, engaging, and accessible introduction to the subject, fully explaining what spin glasses are, why they are important, and how they are opening up new ways of thinking about complexity. This one-of-a-kind guide to spin glasses begins by explaining the fundamentals of order and symmetry in condensed matter physics and how spin glasses fit into and modify this framework. The book then explores how spin-glass concepts and ideas have found applications in areas as diverse as computational complexity, biological and artificial neural networks, protein folding, immune response maturation, combinatorial optimization, and social network modeling. Providing an essential overview of the history, science, and growing significance of this exciting field, the book also features a forward-looking discussion of what spin glasses may teach us in the future about complex systems. This is a useful book for students and practitioners in the natural and social sciences, with new material even for the experts.

Spin glasses are disordered magnetic materials, and it is hard to find a less promising candidate to serve as a focal point of complexity studies, much less as the object of thousands of investigations. On first inspection, they don't seem particularly exciting. Although they're a type of magnet, they're not very good at being magnetic. Metallic spin glasses are unremarkable conductors, and insulating spin glasses are fairly useless as practical insulators. This introductory chapter provides an overview of why spin glasses might be of interest to the reader if they are not a physicist but are interested in any of a variety of other problems outside physics, or more generally in the field of complexity itself. It explores those features of spin glasses that have attracted, in turn, condensed matter and statistical physicists, complexity scientists, and mathematicians and applied mathematicians of various sorts.