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

http://cds.cern.ch/record/348731/files/9803315.pdf

The Hierarchy problem and new dimensions at a millimeter

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

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Quantum field theory and the Jones polynomial

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

Light-cone analysis of virtual processes.

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Topological effects on the physics of the standard model

Noncommuting spatial coordinates and fields can be realized in actual physical situations. Plane wave solutions to noncommuting photodynamics exhibit violaton of Lorentz invariance (special relativity).

Physical Instances of Noncommuting Coordinates

We consider a description of membranes by (2,1)-dimensional field theory, or alternatively a description of irrotational, isentropic fluid motion by a field theory in any dimension. We show that these Galileo-invariant systems, as well as others related to them, admit a peculiar diffeomorphism symmetry, where the transformation rule for coordinates involves the fields. The symmetry algebra coincides with that of the Poincare group in one higher dimension. Therefore, these models provide a nonlinear representation for a dynamical Poincare group.

Field-dependent diffeomorphism symmetry in diverse dynamical systems

Covariant perturbation methods, developed over a quarter century ago, have since then dominated dynamical calculations in quantum field theory. Marvelously successful results can be achieved with some problems (especially in electrodynamics); but in many other contexts the technique yields no information. When the coupling constant is large, perturbation theory is useless. Even for weak coupling there remain phenomena, apparently of physical interest, which cannot easily be seen in the perturbative expansion; for example spontaneous symmetry violation, bound states, entrapment of various excitations. These cooperative, coherent effects can only be exposed by approximation procedures which do not rely on analyticity or regularity in the coupling constant. Such approximations are widely used in quantum mechanics (one does not find the properties of a complex atom or nucleus in the Born series!) and it is profitable to extend them to relativistic quantum field theory.

Roman Jackiw

Papers

Light-cone analysis of virtual processes.

Topological effects on the physics of the standard model

Physical Instances of Noncommuting Coordinates

Field-dependent diffeomorphism symmetry in diverse dynamical systems

Quantum Mechanical Approximations in Quantum Field Theory