Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.

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Topological insulators and superconductors

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

We review the field of cavity optomechanics, which explores the interaction between electromagnetic radiation and nano- or micromechanical motion This review covers the basics of optical cavities and mechanical resonators, their mutual optomechanical interaction mediated by the radiation pressure force, the large variety of experimental systems which exhibit this interaction, optical measurements of mechanical motion, dynamical backaction amplification and cooling, nonlinear dynamics, multimode optomechanics, and proposals for future cavity quantum optomechanics experiments In addition, we describe the perspectives for fundamental quantum physics and for possible applications of optomechanical devices

Cavity Optomechanics

Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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Entanglement in many-body systems

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

The conductance of a sample scattering elastically and coupled to leads with many channels is derived. We assume that all the incident channels on one side of the sample are fed from the same chemical potential. The transmitted and reflected streams are determined by the incident streams through the multichannel scattering properties of the sample. We do not assume that the channels equilibrate with each other. Our result differs from those given earlier by other authors, except for that of Azbel [J. Phys. C 14, L225 (1981)], which is confirmed. We point out that a similar result is obtained for the conductance in a single channel at a temperature above zero. As an application, we obtain the dependence on channel number N of the contributions to the conductance of a small ring, periodic in the Aharonov-Bohm flux through it. Terms whose period is h/e as well as those with period h/2e vary with N as 1/N.

Generalized many-channel conductance formula with application to small rings.

A conductance formula for a sample of arbitrary shape with four terminals is derived to describe transport in the limit where carriers can traverse the sample without suffering phase-destroying events. The Onsager-Casimir symmetry relations are deduced. Experiments measure an off-diagonal Onsager coefficient and the magnetoconductance of such a sample is asymmetric even in the presence of an Aharonov-Bohm flux only. Symmetry relations between conductance measurements which exchange the role of current and voltage leads are predicted.

Four-terminal phase-coherent conductance.

http://ltl.tkk.fi/ice/Kevo/noisereview.pdf

Shot noise in mesoscopic conductors

Under certain conditions, high magnetic fields in a two-dimensional conductor lead to a suppression of both elastic and inelastic backscattering. This, together with the formation of edge states, is used to develop a picture of the integer quantum Hall effect in open multiprobe conductors. We consider both ideal contacts without elastic scattering and also disordered contacts. Ideal contacts populate edge states equally whereas disordered contacts lead to an initial nonequilibrium population of the edge states. In Hall samples much larger than an inelastic length, and in the presence of disordered contacts, the sample edges become equipotential lines only an inelastic scattering length away from the current source and current drain contacts. Samples so small that the carriers can travel from one contact to the other without inelastic relaxation do not exhibit exact quantization if the contacts are disordered. In all cases we find that the quantum Hall effect occurs only if the sample exhibits at least two sets of equilibrated edge states which do not interact via elastic or inelastic scattering. The onset of interaction between the two sets of edge states leads to deviations from exact quantization and eventually to a breakdown of the quantum Hall effect.

Markus Büttiker

Papers

Generalized many-channel conductance formula with application to small rings.

Four-terminal phase-coherent conductance.

Shot noise in mesoscopic conductors

Absence of backscattering in the quantum Hall effect in multiprobe conductors

Josephson behavior in small normal one-dimensional rings