HOME

TheInfoList



OR:

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
order is a kind of order in the zero-temperature
phase of matter In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetizati ...
(also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian
geometric phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Ha ...
s of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition. Various topologically ordered states have interesting properties, such as (1)
topological degeneracy In quantum mechanics, quantum many-body physics, topological degeneracy is a phenomenon in which the ground state of a gapped Hamiltonian, gapped many-body Hamiltonian becomes degenerate in the thermodynamic limit, limit of large system size such t ...
and
fractional statistics In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...
or non-abelian statistics that can be used to realize a topological quantum computer; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi statistics that suggest a quantum information origin of elementary particles; See also (4)
topological entanglement entropy The topological entanglement entropy or ''topological entropy'', usually denoted by \gamma, is a number characterizing many-body states that possess topological order. A non-zero topological entanglement entropy reflects the presence of long ran ...
that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as
spin liquid In condensed matter physics, a quantum spin liquid is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglem ...
s and the
quantum Hall effect The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exh ...
, along with potential applications to fault-tolerant quantum computation.
Topological insulators A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an ...
and topological superconductors (beyond 1D) do not have topological order as defined above, their entanglements being only short-ranged.


Background

Matter composed of
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...
s can have different properties and appear in different forms, such as
solid Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
,
liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
,
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
, etc. These various forms of matter are often called
states of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Many intermediate states are known to exist, such as liquid crystal, ...
or phases. According to
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
and the principle of
emergence In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
, the different properties of materials generally from the different ways in which the atoms are organized in the materials. Those different organizations of the atoms (or other particles) are formally called the
orders Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
in the materials. Atoms can organize in many ways which lead to many different orders and many different types of materials. Landau symmetry-breaking theory provides a general understanding of these different orders. It points out that different orders really correspond to different symmetries in the organizations of the constituent atoms. As a material changes from one order to another order (i.e., as the material undergoes a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
), what happens is that the symmetry of the organization of the atoms changes. For example, atoms have a random distribution in a
liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
, so a liquid remains the same as we displace atoms by an arbitrary distance. We say that a liquid has a ''continuous translation symmetry''. After a phase transition, a liquid can turn into a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
. In a crystal, atoms organize into a regular array (a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
). A lattice remains unchanged only when we displace it by a particular distance (integer times a
lattice constant A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal. A simple cubic crystal has o ...
), so a crystal has only ''discrete translation symmetry''. The phase transition between a liquid and a crystal is a transition that reduces the continuous translation symmetry of the liquid to the discrete symmetry of the crystal. Such a change in symmetry is called ''symmetry breaking''. The essence of the difference between liquids and crystals is therefore that the organizations of atoms have different symmetries in the two phases. Landau symmetry-breaking theory has been a very successful theory. For a long time, physicists believed that Landau Theory described all possible orders in materials, and all possible (continuous) phase transitions.


Discovery and characterization

However, since the late 1980s, it has become gradually apparent that Landau symmetry-breaking theory may not describe all possible orders. In an attempt to explain
high temperature superconductivity High-temperature superconductors (abbreviated high-c or HTS) are defined as materials that behave as superconductors at temperatures above , the boiling point of liquid nitrogen. The adjective "high temperature" is only in respect to previo ...
the
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
spin state was introduced. At first, physicists still wanted to use Landau symmetry-breaking theory to describe the chiral spin state. They identified the chiral spin state as a state that breaks the time reversal and parity symmetries, but not the spin rotation symmetry. This should be the end of the story according to Landau's symmetry breaking description of orders. However, it was quickly realized that there are many different chiral spin states that have exactly the same symmetry, so symmetry alone was not enough to characterize different chiral spin states. This means that the chiral spin states contain a new kind of order that is beyond the usual symmetry description.
Xiao-Gang Wen Xiao-Gang Wen (; born November 26, 1961) is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theor ...
, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces"
The proposed, new kind of order was named "topological order". The name "topological order" is motivated by the low energy
effective theory In science, an effective theory is a scientific theory which proposes to describe a certain set of observations, but explicitly without the claim or implication that the mechanism employed in the theory has a direct counterpart in the actual causes ...
of the chiral spin states which is a
topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...
(TQFT).Atiyah, Michael (1988), "Topological quantum field theories", Publications Mathe'matiques de l'IHéS (68): 175, , , http://www.numdam.org/item?id=PMIHES_1988__68__175_0Witten, Edward (1988), "Topological quantum field theory", ''Communications in Mathematical Physics'' 117 (3): 353, , , http://projecteuclid.org/euclid.cmp/1104161738 New quantum numbers, such as ground state degeneracy (which can be defined on a closed space or an open space with gapped boundaries, including both Abelian topological orders and non-Abelian topological orders) and the non-Abelian
geometric phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Ha ...
of degenerate ground states, were introduced to characterize and define the different topological orders in chiral spin states. More recently, it was shown that topological orders can also be characterized by
topological entropy In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
. But experiments soon indicated that chiral spin states do not describe high-temperature superconductors, and the theory of topological order became a theory with no experimental realization. However, the similarity between chiral spin states and quantum Hall states allows one to use the theory of topological order to describe different quantum Hall states. Just like chiral spin states, different quantum Hall states all have the same symmetry and are outside the Landau symmetry-breaking description. One finds that the different orders in different quantum Hall states can indeed be described by topological orders, so the topological order does have experimental realizations. The fractional quantum Hall (FQH) state was discovered in 1982 before the introduction of the concept of topological order in 1989. But the FQH state is not the first experimentally discovered topologically ordered state. The superconductor, discovered in 1911, is the first experimentally discovered topologically ordered state; it has Z2 topological order. Although topologically ordered states usually appear in strongly interacting boson/fermion systems, a simple kind of topological order can also appear in free fermion systems. This kind of topological order corresponds to integral quantum Hall state, which can be characterized by the
Chern number In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
of the filled energy band if we consider integer quantum Hall state on a lattice. Theoretical calculations have proposed that such Chern numbers can be measured for a free fermion system experimentally. We note that the Chern number of a filled band can only characterize a particular kind of topological order -- integral quantum Hall state. The Chern number and the above proposed experiments cannot probe more generic topological orders, such as the ''Z''2 topological order. Because of this, it is not proper to put the measurement of Chern number at the beginning of this article. <--> It is also well known that such a Chern number can be measured (maybe indirectly) by edge states. The most important characterization of topological orders would be the underlying fractionalized excitations (such as
anyons In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...
) and their fusion statistics and braiding statistics (which can go beyond the quantum statistics of
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
or
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
). Current research works show that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the crucial signatures for identifying 3+1 dimensional topological orders. The multi-loop/string-braiding statistics of 3+1 dimensional topological orders can be captured by the link invariants of particular
topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...
in 4 spacetime dimensions.


Mechanism

A large class of 2+1D topological orders is realized through a mechanism called string-net condensation. This class of topological orders can have a gapped edge and are classified by unitary fusion category (or
monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
) theory. One finds that string-net condensation can generate infinitely many different types of topological orders, which may indicate that there are many different new types of materials remaining to be discovered. The collective motions of condensed strings give rise to excitations above the string-net condensed states. Those excitations turn out to be
gauge bosons In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge ...
. The ends of strings are defects which correspond to another type of excitations. Those excitations are the gauge charges and can carry
Fermi Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...
or
fractional statistics In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...
. The condensations of other extended objects such as "
membranes A membrane is a selective barrier; it allows some things to pass through but stops others. Such things may be molecules, ions, or other small particles. Membranes can be generally classified into synthetic membranes and biological membranes. Bi ...
", "brane-nets", and
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
also lead to topologically ordered phases and "quantum glassiness".


Mathematical formulation

We know that group theory is the mathematical foundation of symmetry-breaking orders. What is the mathematical foundation of topological order? It was found that a subclass of 2+1D topological orders—Abelian topological orders—can be classified by a K-matrix approach. The string-net condensation suggests that tensor category (such as
fusion category In mathematics, a fusion category is a category that is rigid, semisimple, k-linear, monoidal and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If the ground field k is algebraically closed ...
or
monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
) is part of the mathematical foundation of topological order in 2+1D. The more recent researches suggest that (up to invertible topological orders that have no fractionalized excitations): * 2+1D bosonic topological orders are classified by unitary modular tensor categories. * 2+1D bosonic topological orders with symmetry G are classified by G-crossed tensor categories. * 2+1D bosonic/fermionic topological orders with symmetry G are classified by unitary braided fusion categories over symmetric fusion category, that has modular extensions. The symmetric fusion category Rep(G) for bosonic systems and sRep(G) for fermionic systems. Topological order in higher dimensions may be related to n-Category theory. Quantum
operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of ...
is a very important mathematical tool in studying topological orders. Some also suggest that topological order is mathematically described by ''extended quantum symmetry''.


Applications

<--> The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example,
ferromagnet Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...
ic materials that break
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
rotation symmetry can be used as the media of digital information storage. A hard drive made of ferromagnetic materials can store
gigabyte The gigabyte () is a multiple of the unit byte for digital information. The prefix ''giga'' means 109 in the International System of Units (SI). Therefore, one gigabyte is one billion bytes. The unit symbol for the gigabyte is GB. This defini ...
s of information.
Liquid crystal Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. T ...
s that break the rotational symmetry of
molecules A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
find wide application in display technology. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make
semiconducting A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
devices such as
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ...
s. Different types of topological orders are even richer than different types of symmetry-breaking orders. This suggests their potential for exciting, novel applications. One theorized application would be to use topologically ordered states as media for
quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
in a technique known as
topological quantum computing A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to fo ...
. A topologically ordered state is a state with complicated non-local
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
. The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles. As a result, the pattern of quantum entanglements cannot be destroyed by local perturbations. This significantly reduces the effect of
decoherence Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
. This suggests that if we use different quantum entanglements in a topologically ordered state to encode quantum information, the information may last much longer. The quantum information encoded by the topological quantum entanglements can also be manipulated by dragging the topological defects around each other. This process may provide a physical apparatus for performing
quantum computation Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
s. Therefore, topologically ordered states may provide natural media for both
quantum memory In quantum computing, quantum memory is the quantum-mechanical version of ordinary computer memory. Whereas ordinary memory stores information as binary states (represented by "1"s and "0"s), quantum memory stores a quantum state for later ...
and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made
fault tolerant Fault tolerance is the property that enables a system to continue operating properly in the event of the failure of one or more faults within some of its components. If its operating quality decreases at all, the decrease is proportional to the ...
. Topologically ordered states in general have a special property that they contain non-trivial boundary states. In many cases, those boundary states become perfect conducting channel that can conduct electricity without generating heat. This can be another potential application of topological order in electronic devices. Some one put a picture of topological insulator (which has no topological order) in this page. So we have to clarify that topological insulator is not an example of topological order. Topological insulator is an example another kind of order called SPT order. <--> Similarly to topological order,
topological insulator A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an ...
s also have gapless boundary states. The boundary states of topological insulators play a key role in the detection and the application of topological insulators. This observation naturally leads to a question: are topological insulators examples of topologically ordered states? In fact
topological insulator A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an ...
s are different from topologically ordered states defined in this article.
Topological insulator A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an ...
s only have short-ranged entanglements and have no topological order, while the topological order defined in this article is a pattern of long-range entanglement. Topological order is robust against any perturbations. It has emergent gauge theory, emergent fractional charge and fractional statistics. In contrast, topological insulators are robust only against perturbations that respect time-reversal and U(1) symmetries. Their quasi-particle excitations have no fractional charge and fractional statistics. Strictly speaking, topological insulator is an example of symmetry-protected topological (SPT) order, where the first example of SPT order is the Haldane phase of spin-1 chain. But the Haldane phase of spin-2 chain has no SPT order.


Potential impact

Landau symmetry-breaking theory is a cornerstone of
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
. It is used to define the territory of condensed matter research. The existence of topological order appears to indicate that nature is much richer than Landau symmetry-breaking theory has so far indicated. So topological order opens up a new direction in condensed matter physics—a new direction of highly entangled quantum matter. We realize that quantum phases of matter (i.e. the zero-temperature phases of matter) can be divided into two classes: long range entangled states and short range entangled states. Topological order is the notion that describes the long range entangled states: topological order = pattern of long range entanglements. Short range entangled states are trivial in the sense that they all belong to one phase. However, in the presence of symmetry, even short range entangled states are nontrivial and can belong to different phases. Those phases are said to contain SPT order. SPT order generalizes the notion of topological insulator to interacting systems. Some suggest that topological order (or more precisely, string-net condensation) in local bosonic (spin) models have the potential to provide a unified origin for
photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alway ...
,
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
and other
elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiqu ...
in our universe.


See also

*
AKLT model The AKLT model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Ian Affleck, Elliott H. Lieb, Tom Kennedy and provided crucial insight into the physics of the spin-1 Heise ...
*
Fractionalization In quantum mechanics, fractionalization is the phenomenon whereby the quasiparticles of a system cannot be constructed as combinations of its elementary constituents. One of the earliest and most prominent examples is the fractional quantum Hall ef ...
*
Herbertsmithite Herbertsmithite is a mineral with chemical structure Zn Cu3( OH)6 Cl2. It is named after the mineralogist Herbert Smith (1872–1953) and was first found in 1972 in Chile. It is polymorphous with kapellasite and closely related to paratacamit ...
* Implicate order *
Quantum topology Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology. Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associa ...
*
Spin liquid In condensed matter physics, a quantum spin liquid is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglem ...
*
String-net liquid In condensed matter physics, a string-net is an extended object whose collective behavior has been proposed as a physical mechanism for topological order by Michael A. Levin and Xiao-Gang Wen. A particular string-net model may involve only clos ...
* Symmetry-protected topological order *
Topological defect A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological ...
*
Topological degeneracy In quantum mechanics, quantum many-body physics, topological degeneracy is a phenomenon in which the ground state of a gapped Hamiltonian, gapped many-body Hamiltonian becomes degenerate in the thermodynamic limit, limit of large system size such t ...
*
Topological entropy in physics The topological entanglement entropy or ''topological entropy'', usually denoted by \gamma, is a number characterizing many-body states that possess topological order. A non-zero topological entanglement entropy reflects the presence of long ran ...
*
Topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...
*
Topological quantum number In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are ...
*
Topological string theory In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological qu ...


Notes


References


References by categories


Fractional quantum Hall states

* *


Chiral spin states

* *


Early characterization of FQH states

* Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett., 58, 1252 (1987) * Effective-Field-Theory Model for the Fractional Quantum Hall Effect, S. C. Zhang and T. H. Hansson and S. Kivelson, Phys. Rev. Lett., 62, 82 (1989)


Topological order

*
Xiao-Gang Wen Xiao-Gang Wen (; born November 26, 1961) is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theor ...
, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces" * *
Xiao-Gang Wen Xiao-Gang Wen (; born November 26, 1961) is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theor ...
, ''Quantum Field Theory of Many Body Systems – From the Origin of Sound to an Origin of Light and Electrons'', Oxford Univ. Press, Oxford, 2004.


Characterization of topological order

* D. Arovas and J. R. Schrieffer and F. Wilczek, Phys. Rev. Lett., 53, 722 (1984), "Fractional Statistics and the Quantum Hall Effect" * * * *


Effective theory of topological order

*


Mechanism of topological order

* * * *


Quantum computing

* Chetan Nayak, Steven H. Simon, Ady Stern,
Michael Freedman Michael Hartley Freedman (born April 21, 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gene ...
, Sankar Das Sarma, http://www.arxiv.org/abs/0707.1889, 2007, "Non-Abelian Anyons and Topological Quantum Computation", Rev. Mod. Phys. 80, 1083 (2008). * * * * Ady Stern and Bertrand I. Halperin, Phys. Rev. Lett., 96, 016802 (2006), Proposed Experiments to probe the Non-Abelian nu=5/2 Quantum Hall State


Emergence of elementary particles

*
Xiao-Gang Wen Xiao-Gang Wen (; born November 26, 1961) is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theor ...
, Phys. Rev. D68, 024501 (2003), Quantum order from string-net condensations and origin of light and massless fermions * * See also * Zheng-Cheng Gu and
Xiao-Gang Wen Xiao-Gang Wen (; born November 26, 1961) is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theor ...
, gr-qc/0606100, A lattice bosonic model as a quantum theory of gravity,


Quantum operator algebra

* * Landsman N. P. and Ramazan B., Quantization of Poisson algebras associated to Lie algebroids, in ''Proc. Conf. on Groupoids in Physics, Analysis and Geometry''(Boulder CO, 1999)', Editors J. Kaminker et al.,159{192 Contemp. Math. 282, Amer. Math. Soc., Providence RI, 2001, (also ''math{ph/001005''.)
Non-Abelian Quantum Algebraic Topology (NAQAT) 20 Nov. (2008),87 pages, Baianu, I.C.
* Levin A. and Olshanetsky M., Hamiltonian Algebroids and deformations of complex structures on Riemann curves, ''hep-th/0301078v1.'' * Xiao-Gang Wen, Yong-Shi Wu and Y. Hatsugai., Chiral operator product algebra and edge excitations of a FQH droplet (pdf),''Nucl. Phys. B422'', 476 (1994): Used chiral operator product algebra to construct the bulk wave function, characterize the topological orders and calculate the edge states for some non-Abelian FQH states. * Xiao-Gang Wen and Yong-Shi Wu., Chiral operator product algebra hidden in certain FQH states (pdf),''Nucl. Phys. B419'', 455 (1994): Demonstrated that non-Abelian topological orders are closely related to chiral operator product algebra (instead of conformal field theory).
Non-Abelian theory.
* . * R. Brown, P.J. Higgins, P. J. and R. Sivera, "Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids" ''EMS Tracts in Mathematics'' Vol 15 (2011),

* ttp://planetphysics.org/encyclopedia/QuantumAlgebraicTopologyTopics.html Quantum Algebraic Topology (QAT){dead link, date=January 2018 , bot=InternetArchiveBot , fix-attempted=yes Quantum phases Condensed matter physics Statistical mechanics