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.
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.
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.
The notions of a topological phase and topological order were first introduced in the studies of integer and fractional quantum Hall effects, and further developed in the study of topological insulators and topological superconductors in the past decade. Topological concepts are now widely used in many branches of physics, not only limited to condensed matter systems but also in ultracold atomic systems, photonic materials and trapped ions. Papers published in this focus issue are direct testaments of that, and readers will gain a global view of how topology impacts different branches of contemporary physics. We hope that these pages will inspire new ideas through communication between different fields.
The classification of distinct phases of matter is a central theme in condensed matter physics. A phase of matter can be viewed as an equivalence class of physical states that share a certain set of properties. In the case of topological quantum matter, these properties are associated with topological invariants that take specific values for each equivalence class. Perturbing a gapped state that corresponds to a given topological phase (i.e., smoothly deforming the Hamiltonian of the system) does not affect the topological invariant, as long as the bulk energy gap remains finite. The value of this invariant changes only if the system undergoes a quantum phase transition, which is signaled by the collapse of the gap. Full topological protection can only emerge in interacting systems and requires the presence of long-range entanglement, which is at the origin of intrinsic topological order. However, topological properties protected by symmetries emerge even in noninteracting systems and, perhaps surprisingly, can be described within the framework of good old band theory of solids. Systems belonging to this category are generically called topological insulators, after the class of materials that is largely responsible for the recent surge in research on topological matter. In addition, topological band theory considerations can also be applied to superconductors (and superfluids), as described at the mean-field level using Bogoliubov–de Gennes theory. This chapter summarizes the key ideas behind the topological classification of noninteracting systems and discusses the basic properties of various classes of topological insulators and superconductors. For in-depth discussions and related developments the reader is referred to a number of books [ 54 , 175 , 469 ] and review papers [ 23 , 239 , 240 , 421 ] specifically dedicated to this subject.
By using concepts of topology in mathematics, quantum mechanics, and their synergetic development during the past few decades, condensed matter physicists have discovered new phases of matter and introduced general frameworks to classify them. The research includes a vast gamut from chemistry of atomic orbitals to material science, promising new applications in the growing technologies. This review article aims to provide a better understanding of these unprecedented electron systems and their underlying topological principles. The article consist of two parts. First, there is a historical review of using topological concepts in condensed matter systems. Then, in the second part, we elaborate on some basics of topology in quantum mechanics and the concept of topological invariants.
T HE CLASSIFICATION OF DISTINCT PHASES OF MATTER is a central theme in condensed matter physics. A phase of matter can be viewed as an equivalence class of physical states that share a certain set of properties. In the case of topological quantum matter, these properties are associated with topological invariants that take specific values for each equivalence class. Perturbing a gapped state that corresponds to a given topological phase (i.e., smoothly deforming the Hamiltonian of the system) does not affect the topological invariant, as long as the energy gap remains finite. The value of this invariant changes only if the system undergoes a quantum phase transition, which is signaled by the collapse of the gap. Full topological protection can only emerge in interacting systems and requires the presence of long-range entanglement, which is at the origin of intrinsic topological order. However, topological properties protected by symmetries emerge even in noninteracting systems and, perhaps surprisingly, can be described within the framework of good old band theory of solids. Systems belonging to this category are generically called topological insulators, after the class of materials that is largely responsible for the recent surge in research activity on topological matter. In addition, topological band theory considerations can also be applied to superconductors (and superfluids), as described at the mean-field level using Bogoliubov-de Gennes theory. This chapter summaries the key ideas behind the topological classification of noninteracting systems and discusses the basic properties of various classes of topological insulators and superconductors. For in-depth discussions and related developments the reader is referred to a number of books [48, 402, 153] and review papers [211, 212, 357, 22] specifically dedicated to this subject.
Topology is woven into the subtle fabric of quantum theory. Topological concepts and ideas developed starting in the 1980s by the high energy physics and quantum gravity communities, first viewed by many as nothing but aesthetically appealing abstract possibilities, are central components of modern condensed matter physics and play a key role in the field of quantum information and quantum computation. The discovery of integer and fractional quantum Hall effects and, more recently, of topological insulators and superconductors has brought topological quantum matter in the laboratory, with the perspective of realizing functional platforms for topological quantum computation. Rooted in a solid mathematical formalism and spanning its relevance from fundamental questions in quantum gravity and quantum information to practical aspects regarding new materials and platforms for quantum computation, quantum topology is positioned at a fertile confluence of mathematics, physics, and computer and information science. This book focuses on the aspects of quantum topology that are most relevant from the perspective of condensed matter physics and its relation to quantum computation. We start our journey by sketching the relation between topology and quantum physics and summarizing a few basic mathematical concepts that will be used throughout this work. Beyond dry formal definitions, our main goal is to grasp the delicate root that makes the topological quantum world utterly strange and beautiful and follow it into the realm of condensed matter many-body systems.