Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducting states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on $\mathrm{Hg}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum wells are described that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. Experiments on ${\mathrm{Bi}}_{1$-${}x}{\mathrm{Sb}}_{x}$, ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$, and ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.

Weyl physics emerges in the laboratory: Weyl fermions—massless particles with half-integer spin—were once mistakenly thought to describe neutrinos. Although not yet observed among elementary particles, Weyl fermions may exist as collective excitations in so-called Weyl semimetals. These materials have an unusual band structure in which the linearly dispersing valence and conduction bands meet at discrete “Weyl points.” Xu et al. used photoemission spectroscopy to identify TaAs as a Weyl semimetal capable of hosting Weyl fermions. In a complementary study, Lu et al. detected the characteristic Weyl points in a photonic crystal. The observation of Weyl physics may enable the discovery of exotic fundamental phenomena. Science , this issue p. 613 and 622

Recent experiments and theories have suggested that strong spin–orbit coupling effects in certain band insulators can give rise to a new phase of quantum matter, the so-called topological insulator, which can show macroscopic quantum-entanglement effects^{1,2,3,4,5,6,7}. Such systems feature two-dimensional surface states whose electrodynamic properties are described not by the conventional Maxwell equations but rather by an attached axion field, originally proposed to describe interacting quarks^{8,9,10,11,12,13,14,15}. It has been proposed that a topological insulator² with a single Dirac cone interfaced with a superconductor can form the most elementary unit for performing fault-tolerant quantum computation¹⁴. Here we present an angle-resolved photoemission spectroscopy study that reveals the first observation of such a topological state of matter featuring a single surface Dirac cone realized in the naturally occurring Bi₂Se₃ class of materials. Our results, supported by our theoretical calculations, demonstrate that undoped Bi₂Se₃ can serve as the parent matrix compound for the long-sought topological device where in-plane carrier transport would have a purely quantum topological origin. Our study further suggests that the undoped compound reached via n-to-p doping should show topological transport phenomena even at room temperature.

Weyl fermions are massless chiral fermions that play an important role in quantum field theory but have never been observed as fundamental particles. A Weyl semimetal is an unusual crystal that hosts Weyl fermions as quasiparticle excitations and features Fermi arcs on its surface. Such a semimetal not only provides a condensed matter realization of the anomalies in quantum field theories but also demonstrates the topological classification beyond the gapped topological insulators. Here, we identify a topological Weyl semimetal state in the transition metal monopnictide materials class. Our first-principles calculations on TaAs reveal its bulk Weyl fermion cones and surface Fermi arcs. Our results show that in the TaAs-type materials the Weyl semimetal state does not depend on fine-tuning of chemical composition or magnetic order, which opens the door for the experimental realization of Weyl semimetals and Fermi arc surface states in real materials.

Symmetry-broken three-dimensional (3D) topological Dirac semimetal systems with strong spin-orbit coupling can host many exotic Hall-like phenomena and Weyl fermion quantum transport. Here, using high-resolution angle-resolved photoemission spectroscopy, we performed systematic electronic structure studies on Cd₃As₂, which has been predicted to be the parent material, from which many unusual topological phases can be derived. We observe a highly linear bulk band crossing to form a 3D dispersive Dirac cone projected at the Brillouin zone centre by studying the (001)-cleaved surface. Remarkably, an unusually high in-plane Fermi velocity up to 1.5 × 10⁶ ms⁻¹ is observed in our samples, where the mobility is known up to 40,000 cm² V⁻¹s⁻¹, suggesting that Cd₃As₂ can be a promising candidate as an anisotropic-hypercone (three-dimensional) high spin-orbit analogue of 3D graphene. Our discovery of the Dirac-like bulk topological semimetal phase in Cd₃As₂ opens the door for exploring higher dimensional spin-orbit Dirac physics in a real material.

We show that the strongly spin-orbit coupled materials ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ and their derivatives belong to the $Z_2$ topological-insulator class. Using a combination of first-principles theoretical calculations and photoemission spectroscopy, we directly show that ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ is a large spin-orbit-induced indirect bulk band gap ($\delta \sim 150\mathrm{meV}$) semiconductor whose surface is characterized by a single topological spin-Dirac cone. The electronic structure of self-doped ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ exhibits similar $Z_2$ topological properties. We demonstrate that the dynamics of spin-Dirac fermions can be controlled through systematic Mn doping, making these materials classes potentially suitable for topological device applications.