Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in a variety of solid state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, we demonstrate a re-quantization method that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as a basic ingredient to our understanding of basic material properties.

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Berry phase effects on electronic properties

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.

Topological Photonics

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

Quantum electrodynamical corrections to the fine structure of helium

Both the unusually large magnitude and strong temperature dependence of the extraordinary Hall effect in ferromagnetic materials can be understood as effects of the spin-orbit interaction of polarized conduction electrons. It is shown that the interband matrix elements of the applied electric potential energy combine with the spin-orbit perturbation to give a current perpendicular to both the field and the magnetization. Since the net effect of the spin-orbit interaction is proportional to the extent to which the electron spins are aligned, this current is proportional to the magnetization. The magnitude of the Hall constant is equal to the square of the ordinary resistivity multiplied by functions that are not very sensitive to temperature and impurity content. The experimental results behave in such a way also.

Hall Effect in Ferromagnetics

The investigation of Van Vleck and Weisskopf, on the shape of collision-broadened absorption lines, is extended to high power levels of the exciting radiation. Transitions among the molecular states are then induced at a rate that is not negligible compared with the collision rate, thus invalidating the assumption of thermal equilibrium. The theory is based on a quantum transcription of the previous semiclassical treatment, the essential tool being the density matrix. When the resonant frequencies of the molecule are widely spaced, an absorption line is ultimately broadened as the power level increases. Correspondingly, the peak absorption coefficient decreases, the power absorbed per unit volume approaching saturation with increasing incident power. It is shown that the broadening of an absorption line is not to be attributed to any intrinsic modification of the line shape, but rather to a frequency dependent alteration of the energy level populations. A molecule for which all resonant frequencies coincide, a one-dimensional harmonic oscillator, is treated in Appendix II. No saturation effect occurs here, since the absorption is independent of the molecular distribution among the oscillator energy levels.

A Note on Saturation in Microwave Spectroscopy

The study deals with qualitative features of proportional reasoning on dimensional intensive variables by eleven and thirteen year-old students. These features include types of comparison and types of strategies employed. Four proportional problems, varying in numerical and referential content were administered to 116 eleven year-olds and 137 thirteen year-olds. No significant age effect was found. The relative frequencies with which the type of comparison and various strategies were used was greatly affected by context, numerical content of the problem, and the immediately preceeding task. No type of comparison appeared to be more ‘natural’ than another, as suggested in previous research.

Early adolescents' proportional reasoning on ‘rate’ problems

A New Look at Elementary School Science; Science Curriculum Improvement Study

The vertex operator is examined in lowest order perturbation theory. It is found that, as a function of the invariant momentum transfer ${q}^{2}={\mathrm{q}}^{2}\ensuremath{-}{{q}_{0}}^{2}$, it is analytic in a cut plane with the branch point on the negative real axis. A spectral representation (dispersion relation) may therefore be inferred. The threshold of the spectrum depends on the masses of all fields involved unless certain inequalities hold between the masses of the incident and outgoing particles on one hand and the particles in intermediate states on the other; in that case the threshold depends only on the intermediate masses.

Robert Karplus

Papers

Hall Effect in Ferromagnetics

A Note on Saturation in Microwave Spectroscopy

Early adolescents' proportional reasoning on ‘rate’ problems

A New Look at Elementary School Science; Science Curriculum Improvement Study

Spectral Representations in Perturbation Theory. I. Vertex Function