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.

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Anomalous hall effect

Random-matrix theories in quantum physics : common concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

The transport properties of disordered solids have been the subject of much work since at least the 1950s, but with a new burst of activity during the 1980s which has survived up to the present day. There have been numerous reviews of a more or less specialized nature. The present review aims to fill the niche for a non-specialized review of this very active area of research. The basic concepts behind the theory are introduced with more detailed sections covering experimental results, one-dimensional localization, scaling theory, weak localization, magnetic field effects and fluctuations.

https://iopscience.iop.org/article/10.1088/0034-4885/56/12/001/pdf

Localization: theory and experiment

Mostafa ADIMY Directeur de Recherches à l’INRIA Dir. de thèse Ionel S. CIUPERCA Mâıtre de Conférence à l’Université Lyon 1 Examinateur Michael C. MACKEY Directeur de Recherche à l’Université Mc GIll Dir. de thèse Sylvie MÉLÉARD Professeur à l’Ecole Polytechnique Examinatrice Sophie MERCIER Professeur à l’Université de Pau et des Pays de l’Adour Rapportrice Laurent PUJO-MENJOUET Mâıtre de Conférence à l’Université Lyon 1 Dir. de thèse Marta TYRAN-KAMIŃSKA Professeur à l’University of Silesia Examinatrice Bernard YCART Professeur à l’Université de Grenoble Rapporteur

Sous la direction de

We present extensive numerical results applying the finite size scaling method to the theory of electrons in disordered systems. A method is developed for studying the localisation length in 1-dimensional systems of finite cross section. By studying these results as a function of cross-section and using scaling ideas, we derive the critical properties of 2-D and 3-D systems. We calculate transport properties as a function of temperature which can then be compared with experiment.

The scaling theory of electrons in disordered solids: additional numerical results

The localization of the electron states and dc-conductivity of the one dimensional Anderson model are investigated with various numerical procedures. It is found that the eigenstates are always exponentially localized and that in the center of the band the localization length is proportional to the inverse square of the disorder. The dc-conductivity, as obtained by using the Kubo-Greenwood formula, obeys the central limit theorem for any finite imaginary frequency, with a variance, which is inversely proportional to the squareroot of the number of states contributing to the transport. There is no exponential length dependence of the Kubo-Greenwood conductivity within this model. The conductivity tends to zero only in the limit of vanishing imaginary frequency.

Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results

Nonvanishing zero temperature static conductivity in one dimensional disordered systems

Thedc-conductivity of the one-dimensional Anderson model of a disordered system is investigated with two independent numerical methods for systems as large as 104 sites starting from Kubo's formula. In contradiction to the conventional theory of one-dimensional disordered systems a non zero conductivity for finite values of the disorder parameterW/V is found. the eigenstates of the system are investigated. Results for the inverse participation number and the mean square extension of the states are given and compared with the results for the conductivity.

B. Kramer

Papers

The scaling theory of electrons in disordered solids: additional numerical results

Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results

Nonvanishing zero temperature static conductivity in one dimensional disordered systems

Numerical investigation of the conductivity and electron localization of a one dimensional disordered system

MacKinnon and Kramer Respond