Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrodinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.

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Two-dimensional gas of massless Dirac fermions in graphene

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 ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$, ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$, 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.

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Colloquium: Topological insulators

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

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Non-Abelian Anyons and Topological Quantum Computation

Anyons in an exactly solved model and beyond

2. GRAVITY 7. MICROSCOPIC PHYSICS 13. TOPOLOGY OF DEFECTS 18. ANOMALOUS NON-CONSERVATION OF FERMIONIC CHARGE 22. EDGE STATES AND FERMION ZERO MODES ON SOLITON 26. LANDAU CRITICAL VELOCITY 29. CASIMIR EFFECT AND VACUUM ENERGY

The Universe in a Helium Droplet

Superconductivity with lines of GAP nodes: density of states in the vortex

TOPOLOGICAL defects formed during a rapid symmetry-breaking phase transition in the early Universe1,2 could be responsible for seeding large-scale structure, for the anisotropy of the microwave background radiation, and for the predominance of matter over antimatter3,4. The theory describing this cosmological phase transition is formally analogous to that describing the transition to the superfluid state in liquid 3He, so that in principle the process of cosmological defect formation can be modelled in the laboratory. Here we report the results of an experiment in which the 'primordial fireball' is mimicked using a neutron-induced nuclear reaction (n + 3He → p + 3He + 0.76 MeV) to heat small regions of superfluid 3He above the superfluid transition temperature. These bubbles of normal liquid cool extremely rapidly, and we find that their transition back to the superfluid state is accompanied by the formation of a random network of vortices (the superfluid analogue of cosmic strings). We monitor the evolution of this defect state by rotating the superfluid sample, allowing vortices to escape from the network and thus be probed individually. Our results provide clear confirmation of the idea that topological defects form at a rapid second-order phase transition, and give quantitative support to the Kibble–Zurek mechanism5,6 of cosmological defect formation.

Vortex formation in neutron-irradiated superfluid 3He as an analogue of cosmological defect formation

We show that the topologically protected flat band emerging on a surface of a nodal fermionic system promotes the surface superconductivity due to an infinitely large density of states associated with the flat band. The critical temperature depends linearly on the pairing interaction and can be thus considerably higher than the exponentially small bulk critical temperature. We discuss an example of surface superconductivity in multilayered graphene with rhombohedral stacking.

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High-temperature surface superconductivity in topological flat-band systems

The energy levels of fermions bound to the vortex core are considered for the general case of chiral superconductors. There are two classes of chiral superconductivity: in the class I superconducting state the axisymmetric singly quantized vortex has the same energy spectrum of bound states as in an s-wave superconductor: E=(n+1/2)ω0, with integral n. In class II the corresponding spectrum is E=nω0 and thus contains a state with exactly zero energy. The effect of a single impurity on the spectrum of bound states is also considered. For class I the spectrum acquires the doubled period ΔE=2ω0 and consists of two equidistant sets of levels, in accordance with A. I. Larkin and Yu. N. Ovchinnikov, Phys. Rev. B 57, 5457 (1998). For the class II states the spectrum is not influenced by a single impurity if the same approximation is applied.

Grigory Volovik

Papers

The Universe in a Helium Droplet

Superconductivity with lines of GAP nodes: density of states in the vortex

Vortex formation in neutron-irradiated superfluid 3He as an analogue of cosmological defect formation

High-temperature surface superconductivity in topological flat-band systems

Fermion zero modes on vortices in chiral superconductors