This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

From the Publisher:
Although it originated as a method to study general dynamical systems, symbolic dynamics is useful in coding for data storage and transmission as well as in linear algebra. Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises.

An Introduction to Symbolic Dynamics and Coding

We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying ‘equilibrium glass transition’. After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, ageing and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.

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Glassy dynamics of kinetically constrained models

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). Regarding regularity, we show that the RG map, defined on a suitable space of interactions (=formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce, and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension d⩾3, these pathologies occur in a full neighborhood {β>β0, ¦h¦<e(β)} of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d⩾2, the pathologies occur at low temperatures for arbitrary magnetic field strength. Pathologies may also occur in the critical region for Ising models in dimension d⩾4. We discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems. In addition, we discuss critically the concept of Gibbs measure, which is at the heart of present-day classical statistical mechanics. We provide a careful, and, we hope, pedagogical, overview of the theory of Gibbsian measures as well as (the less familiar) non-Gibbsian measures, emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations. We give a rather complete catalogue of the known examples of such occurrences. The main message of this paper is that, despite a well-established tradition, Gibbsiannessshould not be taken for granted.

Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory

This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kac interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.

Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction

We consider Ising-spin systems starting from an initial Gibbs measure 1) and evolving under a spin-flip dynamics towards a reversible Gibbs measure mu not equal nu. Both v and mu are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure nuS(t) at time t and show the following: (1) For all nu and mu, nuS (t) is Gibbs for small t. (2) If both nu and mu have a high or infinite temperature, then nuS(t) is Gibbs for all t > 0. (3) If nu has a low non-zero temperature and a zero magnetic field and mu has a high or infinite temperature, then nuS(t) is Gibbs for small t and non-Gibbs for large t. (4) If nu has a low non-zero temperature and a non-zero magnetic field and mu has a high or infinite temperature, then nuS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where mu has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.

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Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

Long-range spin systems with random interactions are considered. A simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape, provided there is no external field. The argument is valid for both classical and quantum spin systems, and can be applied to (a) spins randomly distributed on a lattice and interacting via dipolar interactions; and (b) spin systems with potentials of the formJ(x
1,x
2)/|x
1 -x
2|
 αd
 , where theJ(x
1,x
2) are independent random variables with mean zero,d is the dimension, and α > 1/2. The key to the proof is a (multidimensional) subadditive ergodic theorem. As a corollary we show that, for random ferromagnets, the correlation length is a nonrandom quantity.

The Thermodynamic Limit for Long-Range Random Systems

We present a class of examples of nearest-neighbour, bounded-spin models, in which the low-temperature Gibbs measures do not converge as the temperature is lowered to zero, in any dimension.

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Chaotic temperature dependence at zero temperature.

A simple, exactly soluble, model of a spin-glass with weakly correlated disorder is presented. It includes both randomness and frustration, but its solution can be obtained without replicas. As the temperatureT is lowered, the spin-glass phase is reached via an equilibrium phase transition atT=T

 f
 . The spin-glass magnetization exhibits a distinctS-shape character, which is indicative of a field-induced transition to a state of higher magnetization above a certain threshold field. For suitable probability distributions of the exchange interactions.
 The physical origin of the dependence upon the probability distributions is explained, and a careful analysis of the ground state structure is given.

On a Classical Spin Glass Model

We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-ip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-ip dynamics of the empirical measure of the spins in a large block in Z d satises

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van Aernout Enter

Papers

Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

The Thermodynamic Limit for Long-Range Random Systems

Chaotic temperature dependence at zero temperature.

On a Classical Spin Glass Model

A large-deviation view on dynamical gibbs-non-gibbs transitions