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Spin Glasses: General Features

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

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 ... More

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.Less

Short-Range Spin Glasses: Some Basic Questions

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

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 ... More

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.Less

The Infinite-Range Spin Glass

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

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 ... More

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.Less

Applications to Other Fields

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

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 ... More

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.Less

Are Spin Glasses Complex Systems?

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

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 ... More

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.Less

Why Spin Glasses?

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

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 ... More

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.Less

THEORY OF RANDOM SOLID STATES

M. Mézard

in Stealing the Gold: A celebration of the pioneering physics of Sam Edwards

This chapter is a non-technical, elementary introduction to the theory of glassy phases and their ubiquity. The aim is to provide a guide and some kind of coherent view to the various topics that ... More

This chapter is a non-technical, elementary introduction to the theory of glassy phases and their ubiquity. The aim is to provide a guide and some kind of coherent view to the various topics that have been explored in recent years in this very diverse field, ranging from spin or structural glasses to protein folding, combinatorial optimization, neural networks, error correcting codes, and game theory.Less

Spin glasses

Marc Mézard and Andrea Montanari

in Information, Physics, and Computation

This chapter describes random magnetic systems, ‘spin glasses’, by special random ensembles of factor graphs. It also studies the glass phase, characterized by a freezing of the spins, in the ... More

This chapter describes random magnetic systems, ‘spin glasses’, by special random ensembles of factor graphs. It also studies the glass phase, characterized by a freezing of the spins, in the framework of equilibrium statistical physics. It describes the two types of spin glass phase transitions that have been encountered in mean field models, and introduces the relevant order parameters to describe them based on the notion of overlap. Special attention is given to the concept of frustration — a basic ingredient of spin glasses — which is discussed in conjunction with gauge transformations.Less

REMARKS ON THE EDWARDS–ANDERSON PAPER

P. W. Anderson

in Stealing the Gold: A celebration of the pioneering physics of Sam Edwards

This chapter presents a brief history of the inspirations and ideas behind the famous seminal spin glass paper of Edwards and Anderson, together with some of the further work it stimulated.

EDWARDS–ANDERSON: OPENING UP THE WORLD OF COMPLEXITY

David Sherrington

in Stealing the Gold: A celebration of the pioneering physics of Sam Edwards

This chapter presents a brief review of some of the riches exposed by and others germinated by the seminal spin glass work of Edwards and Anderson. It discusses such things as replica theory, spin ... More

This chapter presents a brief review of some of the riches exposed by and others germinated by the seminal spin glass work of Edwards and Anderson. It discusses such things as replica theory, spin glass dynamics, and probability theory.Less

Special topics and interdisciplinary models

Ralph Skomski

in Simple Models of Magnetism

Atomic disorder has far-reaching consequences for the behavior of magnetic materials. It modifies the electronic structure but does not necessarily destroy ferromagnetism, as exemplified by amorphous ... More

Atomic disorder has far-reaching consequences for the behavior of magnetic materials. It modifies the electronic structure but does not necessarily destroy ferromagnetism, as exemplified by amorphous ferromagnets. Spin glasses combine disorder with competing exchange, and their ground state is neither ferromagnetic nor antiferromagnetic. The equilibrium and nonequilibrium properties of spin glasses have remained a complex problem, and several models have been developed, such as the Edwards–Anderson and Sherrington–Kirkpatrick models. On a mean-field level, the determination of ordering and spin-glass temperatures involves the diagonalisation of large random matrices. This chapter discusses disordered magnets and spin glasses, ferromagnetic order in inhomogeneous magnets, soft matter, transport, magnetism, random walks, polymers, diffusion, polymers and critical dimensionality, gases in magnetic metals, magnetoresistance, Bruggeman model, nanostructures, thin films, length scales in nanomagnetism, random anisotropy, and two-phase nanostructures. The use of models in disciplines other than magnetism is also considered, including metallurgy, biology and medicine, and social sciences.Less

FERROIC MATERIALS

Vinod K. Wadhawan

in Smart Structures: Blurring the Distinction Between the Living and the Nonliving

This chapter discusses ferroic materials, which are the first of the three types of nonlinear-response materials discussed in this book, the other two being soft materials and nanostructured ... More

This chapter discusses ferroic materials, which are the first of the three types of nonlinear-response materials discussed in this book, the other two being soft materials and nanostructured materials. Their nonlinear properties are a consequence of one or more ferroic phase transitions that can occur in them. The chapter begins with an introduction to phase transitions and critical phenomena in crystals. Ferroic phase transitions are those that involve a change of point-group symmetry in a nondisruptive manner. Their thermodynamic classification is described, which helps divide ferroic materials into ferroelectrics, ferromagnetics, ferroelastics, ferrobielastics, ferroelastoelectrics, etc. The poling process for ferroic ceramics is explained. The domain structure of ferroics, and the possibilities of tailoring this structure to advantage are discussed. The importance of multiferroics in smart-structures research is emphasized. There are also sections on spin-glasses, shape-memory alloys and ceramics, and relaxor ferroelectrics. The book has helpful appendices on crystallographic symmetry and on tensor properties.Less

Ising models on random graph

Marc Mézard and Andrea Montanari

in Information, Physics, and Computation

This chapter studies two problems of statistical physics: the ferromagnet and the spin glass, on large random graphs with fixed degree profile. It describes the use of the replica symmetric cavity ... More

This chapter studies two problems of statistical physics: the ferromagnet and the spin glass, on large random graphs with fixed degree profile. It describes the use of the replica symmetric cavity method in this context, and studies its stability. The analysis relies on physicists methods, without any attempt at being rigorous. It provides a complete solution of the ferromagnetic problem at all temperatures. In the spin glass case, the replica symmetric solution is asymptotically correct in the high temperature ‘paramagnetic’ phase, but it turns out to be wrong in the spin glass phase. The phase transition temperature can be computed exactly.Less

Random systems

Hidetoshi Nishimori and Gerardo Ortiz

in Elements of Phase Transitions and Critical Phenomena

Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and ... More

Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and critical phenomena. Although randomness may seem to obscure singular behaviour such as divergence of physical quantities at the critical temperature, it is established that well-defined phase transitions exist as long as randomness is not too strong, but the critical behaviour may get modified with respect to the pure sample. After the introduction of basic concepts and methods such as self-averaging and replica method, it is elucidated what type of phase transitions exist in the random-field Ising model and the SK model of spin glasses. Also explained are the percolation transitions using the fractal structure and the Potts model.Less

 Network Analysis of the Significance of Hippocampal Subfields

Gergely Papp and Alessandro Treves

in Hippocampal Place Fields: Relevance to Learning and Memory

This chapter begins with a discussion of the evolutionary changes in the mammalian nervous system that distinguished it from reptilians and birds. It then discusses differentiation of the hippocampus ... More

This chapter begins with a discussion of the evolutionary changes in the mammalian nervous system that distinguished it from reptilians and birds. It then discusses differentiation of the hippocampus and virtual rat simulations. It argues that hippocampal models require firing-rate adaptation for producing a time-shifted localization, i.e., the prediction of future locations in a spatial environment. The hypothesis is that a major qualitative structural change may have served to produce solely a quantitative functional advantage.Less

REPRINT THEORY OF SPIN GLASSES

S. F. Edwards and P. W. Anderson

in Stealing the Gold: A celebration of the pioneering physics of Sam Edwards

This paper presents a new theory of the class of dilute magnetic alloys, called the spin glasses, which offers a simple explanation of the cusp found experimentally in the susceptibility. The ... More

This paper presents a new theory of the class of dilute magnetic alloys, called the spin glasses, which offers a simple explanation of the cusp found experimentally in the susceptibility. The argument is that because the interaction between the spins dissolved in the matrix oscillates in sign according to distance, there will be no mean ferro- or antiferromagnetism, but there will be a ground state with the spins aligned in definite directions, even in a situation where these directions appear to be at random. At the critical temperature, the existence of these preferred directions affects the orientation of the spins, leading to a cusp in the susceptibility. This cusp is smoothed by an external field.Less