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We present this Collection of research, review and comment from Nature Research to celebrate the award of the 2016 Nobel Prize in Physics to David Thouless, Duncan Haldane and Michael Kosterlitz — who are recognized "for theoretical discoveries of topological phase transitions and topological phases of matter". Once an abstract field of mathematics, topology has become a key ingredient in understanding a variety of phenomena in condensed matter physics.
Using optical lattices to trap ultracold atoms provides a powerful platform for probing topological phases, analogues to those found in condensed matter. But as these systems are highly tunable, they could be used to engineer even more exotic phases.
The quantum Hall effect leads to topologically protected edge states, and for a long time was thought to exclusively emerge in the presence of an external magnetic field. But in 1988, Duncan Haldane proposed a model in which this exotic electronics structure arises without this requirement. He proposed that, in a honeycomb lattice with a staggered flux, the necessary ingredients for a quantum Hall effect would be inherent in the material itself. The principles behind this concept were later recruited to design topological insulators, but in its original expression, the Haldane model has not been observed in the laboratory. In this issue of Nature, two groups report on progress connected to the Haldane model. Gregor Jotzu et al. report the first realization of the Haldane model and Pedram Roushan et al. show how it can be precisely measured. Jotzu et al. use ultracold fermions to realize the breaking of time-reversal and inversion symmetry — the two main requirements of the model — by implementing a circular modulation of the lattice position and an energy offset between neighbouring sites. Roushan et al. use superconducting quantum circuits — a Josephson junction sandwiched between superconducting electrodes — to realize a non-interacting form of the Haldane model with a single qubit and an interacting two-qubit model through a new experimental setup called 'gmon' coupling architecture. Their setup allows them to characterize both cases by measuring the Berry curvature, a feature that all topological structures have in common.
First-principles calculations predict that Bi2Se3, Bi2Te3 and Sb2Te3 are topological insulators—three-dimensional semiconductors with unusual surface states generated by spin–orbit coupling—whose surface states are described by a single gapless Dirac cone. The calculations further predict that Bi2Se3 has a non-trivial energy gap larger than the energy scale kBT at room temperature.
Three-dimensional Dirac semimetals are a recently discovered state of condensed matter considered as the 3D analogue of graphene. Here, Yang et al. propose a general framework to classify stable 3D Dirac semimetals in systems with time-reversal, inversion and uniaxial rotational symmetries.
Physics in a two-dimensional environment is very different from what we observe in the three-dimensional world. If dimensionality is reduced, thermal fluctuations destroy a system's spatial order and most phase transitions, like those responsible for ferromagnetism for instance, cannot occur. But there is a particular type of phase transition, involving the pairing of vortices, that does exist in two dimensions. First predicted 30 years ago by Berezinskii, Kosterlitz and Thouless, the ‘BKT transition’ has now been observed directly for the first time in a planar gas of ultracold rubidium atoms.
When doped with copper, the topological insulator Bi2Se3 becomes superconducting. But for new physics and applications the search is not for just any superconductor; the material must retain its topological character. And indeed that is the case with doped Bi2Se3.
The fractional quantum Hall effect occurs when electrons move in Landau levels. In this study, using a theoretical flat-band lattice model, the fractional quantum Hall effect is observed in the presence of repulsive interactions when the band is one third full and in the absence of Landau levels.
The fabrication of oxide thin-film heterostructures has improved considerably over the past few years. The first demonstration of the fractional quantum Hall effect in an oxide now attests to the potential of these compounds to rival conventional semiconductors.
Non-trivial topological phases can allow for one-way spin-polarized transport along the interfaces of topological insulators but they are relatively uncommon in the condensed state of matter. By arranging judiciously designed metamaterials into two-dimensional superlattices, a photonic topological insulator has now been demonstrated theoretically, enabling unidirectional spin-polarized photon propagation without the application of external magnetic fields or breaking of time-reversal symmetry.
Multi-terminal superconducting Josephson junctions are used to induce topologically protected transitions between gapless and gapped states, showing the potential for creating artificial topological materials.