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

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

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Topological phases have been explored in various fields in physics such as spintronics, photonics, liquid helium, correlated electron system, and cold-atomic system. This leads to the recent foundation of emerging materials such as topological band insulators, topological photonic crystals, and topological superconductors/superfluid. In this paper, we propose a topological magnonic crystal which provides protected chiral edge modes for magnetostatic spin waves. Based on a linearized Landau-Lifshitz equation, we show that a magnonic crystal with the dipolar interaction acquires a spin-wave volume-mode band with nonzero Chern integer. We argue that such magnonic systems are accompanied by the same integer numbers of chiral spin-wave edge modes within a band gap for the volume-mode bands. In these edge modes, the spin wave propagates in a unidirectional manner without being scattered backward, which implements novel fault-tolerant spintronic devices.

Topological chiral magnonic edge mode in a magnonic crystal

The Berry phase due to the spin wave function gives rise to the orbital ferromagnetism and anomalous Hall effect in the noncoplanar antiferromagnetic ordered state on face-centered-cubic (fcc) lattice once the crystal is distorted perpendicular to the $(1,1,1)$ or $(1,1,0)$ plane. The relevance to the real systems $\ensuremath{\gamma}$-FeMn and ${\mathrm{NiS}}_{2}$ is also discussed.

Orbital ferromagnetism and anomalous Hall effect in antiferromagnets on the distorted fcc lattice.

Thermal Hall conductivity of magnons described by a noninteracting boson Hamiltonian is derived by the linear response theory. The thermal Hall conductivity is expressed by the Berry curvature in momentum space, which also has the prevailing form for bosonic systems. This theory covers various spin waves, such as spin waves in antiferromagnets and magnetostatic spin waves. As an example, we calculate the thermal Hall conductivity by the magnetostatic spin wave in yttrium iron garnet and reveal its dependence on a magnetic field and temperature.

Thermal Hall effect of magnons in magnets with dipolar interaction

Based on a linearized Landau-Lifshitz equation, we show that two-dimensional periodic allay of ferromagnetic particles coupled with magnetic dipole-dipole interactions supports chiral spin-wave edge modes, when subjected under the magnetic field applied perpendicular to the plane. The mode propagates along a one-dimensional boundary of the system in a unidirectional way and it always has a chiral dispersion within a band gap for spin-wave volume modes. Contrary to the well-known Damon-Eshbach surface mode, the sense of the rotation depends not only on the direction of the field but also on the strength of the field; its chiral direction is generally determined by the sum of the so-called Chern integers defined for spin-wave volume modes below the band gap. Using simple tight-binding descriptions, we explain how the magnetic dipolar interaction endows spin-wave volume modes with nonzero Chern integers and how their values will be changed by the field.

Chiral spin-wave edge modes in dipolar magnetic thin films

Spin superfluid is a novel emerging quantum matter arising from the Bose-Einstein condensate (BEC) of spin-1 bosons. We demonstrate the spin superfluid ground state in canted antiferromagnetic Cr2O3 thin film at low temperatures via nonlocal spin transport. A large enhancement of the nonlocal spin signal is observed below ~20 K, and it saturates from ~5 down to 2 K. We show that the spins can propagate over very long distances (~20 μm) in such spin superfluid ground state and that the nonlocal spin signal decreases very slowly as the spacing increases with an inverse relationship, which is consistent with theoretical prediction. Furthermore, spin superfluidity has been investigated in the canted antiferromagnetic phase of the (11[Formula: see text]0)-oriented Cr2O3 film, where the magnetic field dependence of the associated critical temperature follows a 2/3 power law near the critical point. The experimental demonstration of the spin superfluid ground state in canted antiferromagnet will be extremely important for the fundamental physics on the BEC of spin-1 bosons and paves the way for future spin supercurrent devices, such as spin-Josephson junctions.

https://advances.sciencemag.org/content/advances/4/4/eaat1098.full.pdf

Ryuichi Shindou

Papers

Topological chiral magnonic edge mode in a magnonic crystal

Orbital ferromagnetism and anomalous Hall effect in antiferromagnets on the distorted fcc lattice.

Thermal Hall effect of magnons in magnets with dipolar interaction

Chiral spin-wave edge modes in dipolar magnetic thin films

Experimental signatures of spin superfluid ground state in canted antiferromagnet Cr2O3 via nonlocal spin transport