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arXiv:1610.03911 · Reviews of Modern Physics

Colloquium : Zoo of quantum-topological phases of matter

Xiao-Gang Wen

Published Dec 4, 2017• 881 citations
arXivHTMLPDF

Abstract

The subject of this Colloquium is related to the topic of the 2016 Physics Nobel Prize that was awarded to David J. Thouless, F. Duncan M. Haldane, and J. Michael Kosterlitz ``for theoretical discoveries of topological phase transitions and topological phases of matter.'' The Colloquium provides a pedagogical introduction to topological phases of matter from comprehensive point of view of many-body entanglement which is important in quantum physics.

DOI
10.1103/revmodphys.89.041004

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Xiao-Gang Wen

What are topological phases of matter? First, they are phases of matter at zero temperature. Second, they have a nonzero energy gap for the excitations above the ground state. Third, they are disordered liquids that seem to have no feature. But those disordered liquids actually can have rich patterns of many-body entanglement representing new kinds of order. This Colloquium gives a simple introduction and a brief survey of topological phases of matter. First topological phases with topological order (i.e., with long-range entanglement) are discussed. Then topological phases without topological order (i.e., with short-range entanglement) are covered.

APS•Dec 1, 2017•70 citations
The subject of this Colloquium is related to the topic of the 2016 Physics Nobel Prize that was awarded to David J. Thouless, F. Duncan M. Haldane, and J. Michael Kosterlitz ``for theoretical discoveries of topological phase transitions and topological phases of matter.'' The Colloquium provides a pedagogical introduction to topological phases of matter from comprehensive point of view of many-body entanglement which is important in quantum physics.
Nobel Lecture, presented December 8, 2016, Aula Magna, Stockholm University. I will describe the history and background of three discoveries cited in this Nobel Prize: The "TKNN" topological formula for the integer quantum Hall effect found by David Thouless and collaborators, the Chern insulator or quantum anomalous Hall effect, and its role in the later discovery of time-reversal-invariant topological insulators, and the unexpected topological spin-liquid state of the spin-1 quantum antiferromagnetic chain, which provided an initial example of topological quantum matter. I will summarize how these early beginnings have led to the exciting, and currently extremely active, field of "topological matter."1 MoreReceived 17 June 2017DOI:https://doi.org/10.1103/RevModPhys.89.040502© 2017 Nobel Foundation, Published by The American Physical Society*The 2016 Nobel Prize for Physics was shared by David J. Thouless, F. Duncan M. Haldane, and John Michael Kosterlitz. These papers are the text of the address given in conjunction with the award.Physics Subject Headings (PhySH)Research AreasTopological defectsTopological phase transitionTopological phases of matterGeneral PhysicsInterdisciplinary Physics
The Nobel Prize in physics of year 2016 has been awarded to David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz for their theoretical discoveries of topological phase transitions and topological phase of matter. This paper starts with a brief introduction to their groundbreaking works made in 1970s–1980s, including the KT phase transition due to topological defects, vortex and antivortex pairs, in two-dimensional superfluid or superconductor, the TKNN number identifying the integer quantum Hall state, as well as the Haldane model to realize quantum Hall effect without external magnetic field in crystal lattice. Their insightful findings have revealed a new quantum state of matter, which cannot be described by local order parameter within Landau symmetry-breaking theory but by some global topological invariant of electronic states. In topology, a branch of mathematics, topological invariant describes the preserved properties of an object under continuous deformations. The combination of topology with the state of matter is a breakthrough in condensed matter physics. The band topology theory based on Berry phase is introduced in order to understand the origin of topology of the electronic states in solids. But only in last ten years, their pioneering works have been rediscovered and greatly extended. Around 2005, the time-reversal symmetry protected topological state identified by Z2 number has been discovered theoretically and soon it is realized in material called as Z2 topological insulator. Quickly, the concept of symmetry protected topological states has been generalized from time-reversal symmetry to all the space group symmetries, from electronic states to those in photonic crystal, phonon spectrum and cold atoms, and even from the insulators to metals. The first experimentally realized topological metal is Dirac semimetal, a three-dimensional version of graphene, with massless Dirac equation describing the low energy excitation around Fermi level. The discovery of Weyl semimetal in TaAs family has stimulated lots of studies on the chiral Weyl fermions, which have not been found in vacuum but observed as quasiparticles in Weyl semimetals. More topological semimetals, such as node-line semimetal and multiple-degenerate nodal point semimetal, have been also proposed. The distinguished property of topological states of matter is its stability in both bulk and boundary states against the perturbation preserving the protecting symmetry. This is expected for potential applications in electronic devices. Therefore, three Laureates of Nobel Prize in physics of 2016 revealed the secrets of exotic matter and opened the door to topological states.
What are topological phases of matter? First, they are phases of matter at zero temperature. Second, they have a nonzero energy gap for the excitations above the ground state. Third, they are disordered liquids that seem to have no feature. But those disordered liquids actually can have rich patterns of many-body entanglement representing new kinds of order. This Colloquium gives a simple introduction and a brief survey of topological phases of matter. First topological phases with topological order (i.e., with long-range entanglement) are discussed. Then topological phases without topological order (i.e., with short-range entanglement) are covered.