There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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The electronic properties of graphene

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

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

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 show how, in principle, to construct analogs of quantum Hall edge states in "photonic crystals" made with nonreciprocal (Faraday-effect) media. These form "one-way waveguides" that allow electromagnetic energy to flow in one direction only.

Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry.

Photonic crystals built with time-reversal-symmetry-breaking Faraday-effect media can exhibit chiral edge modes that propagate unidirectionally along boundaries across which the Faraday axis reverses. These modes are precise analogs of the electronic edge states of quantum-Hall-effect (QHE) systems, and are also immune to backscattering and localization by disorder. The Berry curvature of the photonic bands plays a role analogous to that of the magnetic field in the QHE. Explicit calculations demonstrating the existence of such unidirectionally propagating photonic edge states are presented.

https://arxiv.org/pdf/cond-mat/0602501.pdf

Analogs of quantum-Hall-effect edge states in photonic crystals

We construct time-reversal invariant topological superconductors and superfluids in two and three dimensions. These states have a full pairing gap in the bulk, gapless counterpropagating Majorana states at the boundary, and a pair of Majorana zero modes associated with each vortex. The superfluid $^{3}\mathrm{He}$ $B$ phase provides a physical realization of the topological superfluidity, with experimentally measurable surface states protected by the time-reversal symmetry. We show that the time-reversal symmetry naturally emerges as a supersymmetry, which changes the parity of the fermion number associated with each time-reversal invariant vortex and connects each vortex with its superpartner.

Time-reversal-invariant topological superconductors and superfluids in two and three dimensions.

We consider extended Hubbard models with repulsive interactions on a honeycomb lattice, and the transitions from the semimetal to Mott insulating phases at half-filling. Because of the frustrated nature of the second-neighbor interactions, topological Mott phases displaying the quantum Hall and the quantum spin Hall effects are found for spinless and spin fermion models, respectively. The mean-field phase diagram is presented and the fluctuations are treated within the random phase approximation. Renormalization group analysis shows that these states can be favored over the topologically trivial Mott insulating states.

Topological Mott insulators.

Following the discovery of the Fe-pnictide superconductors, local-density approximation (LDA) band structure calculations showed that the dominant contributions to the spectral weight near the Fermi energy came from the $\text{Fe}\text{ }3d$ orbitals. The Fermi surface is characterized by two hole surfaces around the $\ensuremath{\Gamma}$ point and two electron surfaces around the $M$ point of the two Fe/cell Brillouin zone. Here, we describe a two-band model that reproduces the topology of the LDA Fermi surface and exhibits both ferromagnetic and $q=(\ensuremath{\pi},0)$ spin-density wave fluctuations. We argue that this minimal model contains the essential low energy physics of these materials.

Srinivas Raghu

Papers

Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry.

Analogs of quantum-Hall-effect edge states in photonic crystals

Time-reversal-invariant topological superconductors and superfluids in two and three dimensions.

Topological Mott insulators.

Minimal two-band model of the superconducting iron oxypnictides