The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

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

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE), focusing on recent developments that have provided a more complete framework for understanding this subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergy between experimental and theoretical work, both playing a crucial role, has been at the heart of these advances. On the theoretical front, the adoption of Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents which originate from spin-orbit coupling. On the experimental front, new experimental studies of the AHE in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors, have more clearly established systematic trends. These two developments in concert with first-principles electronic structure calculations, strongly favor the dominance of an intrinsic Berry-phase-related AHE mechanism in metallic ferromagnets with moderate conductivity. The intrinsic AHE can be expressed in terms of Berry-phase curvatures and it is therefore an intrinsic quantum mechanical property of a perfect cyrstal. An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highly conductive ferromagnets. We review the full modern semiclassical treatment of the AHE together with the more rigorous quantum-mechanical treatments based on the Kubo and Keldysh formalisms, taking into account multiband effects, and demonstrate the equivalence of all three linear response theories in the metallic regime. Finally we discuss outstanding issues and avenues for future investigation.

/pdf/anomalous-hall-effect-tyeyld308t.pdf

Anomalous hall effect

This is a tutorial review on the Potts model aimed at bringing out in an organized fashion the essential and important properties of the standard Potts model. Emphasis is placed on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective. Topics reviewed include the mean-field theory, duality relations, series expansions, critical properties, experimental realizations, and the relationship of the Potts model with other lattice-statistical problems.

The Potts model

The dislocation theory of two-dimensional melting due to Kosterlitz and Thouless is investigated for the triangular lattice, paying special attention to angular forces between dislocation pairs, which are equal in magnitude to the radial forces. Generalizing the dislocation Hamiltonian to an arbitrary vector Coulomb gas with different radial and angular interactions we find ${K}_{R}^{i}(T)\ensuremath{-}{K}_{R}^{i}({T}_{c}^{\ensuremath{-}})\ensuremath{\sim}{t}^{\overline{\ensuremath{\nu}}}$, where ${K}_{R}^{i}$ is a renormalized coupling which includes the screening effect of bound dislocation pairs, the superscript $i$ signifies either a radial, $r$, or angular, $\ensuremath{\theta}$ part and $t$ is the reduced temperature, $t=\frac{({T}_{c}\ensuremath{-}T)}{{T}_{c}}$. The exponent $\overline{\ensuremath{\nu}}$ varies as the ratio $\frac{{K}_{R}^{\ensuremath{\theta}}({T}_{c}^{\ensuremath{-}})}{{K}_{R}^{r}({T}_{c}^{\ensuremath{-}})}$ is changed and is equal to 0.3696... for the physical value of ${K}^{\ensuremath{\theta}}={K}^{r}$. In this case the shear and bulk elastic constants have the same temperature dependence as the ${K}_{R}^{i}$. We find that ${K}_{R}^{r}$ has finite universal value at ${T}_{c}$ and ${K}_{R}^{i}=0$ for $Tg{T}_{c}$, corresponding to metallic behavior of the vector Coulomb gas.

Melting and the vector Coulomb gas in two dimensions

Experiments on the random field Ising model, DB Belanger beyond the Sherrington-Kirkpatrick model, C de Dominicis non-equilibrium dynamics, J Kurchan the vortex glass, DJ Bishop and P Gammel vulcanization of rubber, A Zippelius and P Goldbart quadrupolar spin glasses, K Binder theory of the random field Ising model, T Nattermann quantum spin glasses, R Bhatt protein folding, H Orland et al simulations of spin glasses, G Parisi spin glasses without disorder and the similarities between glasses and spin glasses, M Mezard and JP Bouchaud dynamics of lines in disordered media, P Le Doussal experiments of spin glass systems, P Norblad nose from spin fluctuations in spin glasses, M Weissman

Spin glasses and random fields

Avec la motivation d'une relation possible entre la supraconductivite dans les nouveaux oxydes supraconducteurs et le comportement magnetique de ces materiaux, on realise des simulations numeriques pour determiner l'aimantation de sous-reseaux dans l'etat fondamental de l'antiferromagnetique a spin 1/2 sur le reseau carre avec des interactions de plus proches voisins

Monte Carlo simulations of the spin-1/2 Heisenberg antiferromagnet on a square lattice

The three-dimensional Ising spin-glass in zero field with nearest-neighbor interactions having \ifmmode\pm\else\textpm\fi{}J distribution is studied by Monte Carlo simulations for samples of linear dimension L with 3\ensuremath{\le}L\ensuremath{\le}20. Results for the probability distribution, ${\mathrm{P}}_{\mathrm{L}}$(q), of the spin-glass order parameter are analyzed by finite-size scaling. Data for T\ensuremath{\ge}1.2 are consistent with a conventional phase transition at ${\mathrm{T}}_{\mathrm{c}}$${=1.2}_{\mathrm{\ensuremath{-}}0.2}^{+0.1}$, with exponents \ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\eta}=-0.3\ifmmode\pm\else\textpm\fi{}0.2. However, results at lower temperature indicate marginal behavior and suggest that the lower critical dimensional is close to three.

A. P. Young

Papers

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

Melting and the vector Coulomb gas in two dimensions

Spin glasses and random fields

Monte Carlo simulations of the spin-1/2 Heisenberg antiferromagnet on a square lattice

Search for a transition in the three-dimensional J Ising spin-glass