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Reviews of Modern Physics

Nobel Lecture: Topological quantum matter

F. D. M. Haldane

Published Oct 9, 2017• 441 citations
Publisher

Abstract

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

DOI
10.1103/revmodphys.89.040502

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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.
Topology used to be a term confined to a branch of pure mathematics, where it referred to an invariant property of shape. The classic example was the way objects containing a single hole, like a torus and a coffee cup with handle, can be smoothly moulded into one another without tearing. But topological considerations have long played a role in the physics of matter, where for example they might dictate particular arrangements of component parts that can't be erased from the system. The classic example here is the fact that a 'hairy ball' can't be combed flat without having at least two pointy tufts. Such 'defects' in organization can be considered 'topologically protected', since they are robust against any recombing of the hair. They are universal features that don't depend on the material specifics of the system: topological defects in liquid crystals are analogous to defects in spacetime called cosmic strings. In the past several decades in particular, properties of matter arising from topological considerations have become a major theme, reflected for example in the award of the 1985 and 1998 Nobel Prizes in Physics for discoveries involving the quantum Hall effect. Here the 'Hall conductance', quantifying the passage of electrical current in a 2D conductor in the presence of a transverse magnetic field, takes precise integral or fractional multiples of a particular quantized value related to the electron charge. This behaviour persists regardless of how we modify the material, for example by adding impurities. Topological phases and transitions were also a feature of the work that won the 2016 Nobel Prize. It has become recognized that the topological properties of the quantum-mechanical electronic structures of certain materials can give them unusual and perhaps useful properties. Some researchers think, for example, that 'topological matter' might supply quantum bits for quantum computation that resist the randomizing effects of noise. Xiao-Gang Wen of the Massachusetts Institute of Technology has been developing ideas about 'topological order' in fundamental physics for several decades. His notions of how topology in the underlying structure of spacetime might give rise to fundamental particles and forces make a connection to the topological phases recognized in condensed matter, revealing a new unifying principle in physics. National Science Review spoke to him about his work.