Topological order
Encyclopedia
In physics
, topological order is a new kind of order (a new kind of organization of particles) in a quantum state that is beyond the Landau symmetry-breaking
description. It cannot be described by local order parameters
and long range correlations. However, topological orders can be described by a new set of quantum number
s, such as ground state degeneracy, non-Abelian Berry phases of degenerate ground states, quasiparticle
fractional statistics, edge states, topological entanglement entropy, etc. Strictly speaking, topological order is a pattern of long-range quantum entanglement
in quantum states, defined via local unitary transformations. States with different topological orders cannot change into each other without a phase transition.
s, matter can have very different properties and appear in very different forms, such as solid
, liquid
, superfluid
, magnet
, etc. According to condensed matter physics
and the principle of emergence, the different properties of materials originate 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
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
), what happens is that the symmetry of the organization of the atoms changes.
For example, atoms have a random distribution in a liquid
, so a liquid remains the same as we displace it 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
. In a crystal, atoms organize into a regular array (a lattice
). A lattice remains unchanged only when we displace it by a particular distance (integer times of lattice constant
), 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 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 is a very successful theory. For a long time, physicists believed that Landau symmetry-breaking
theory describes all possible orders in materials, and all possible (continuous) phase transitions.
of the chiral spin states which is a topological quantum field theory
(TQFT) ). New quantum numbers, such as ground state
degeneracy
and the non-Abelian
Berry's phase of degenerate ground states . were introduced to characterize the different topological orders in chiral spin states. Recently, it was shown that topological orders can also be characterized by topological entropy
.
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 beyond 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.
Fractional quantum Hall (FQH) states were discovered in 1982 before the introduction of the concept of topological order. But FQH states are not the first experimentally discovered topologically ordered states. Superconductors discovered in 1911 are the first, which have a Z2 topological order (note that superconductivity can be described by the Ginzburg-Landau theory WITH dynamical U(1) gauge field, which is
a Z2 gauge theory --
an effective theory of Z2 topological order. The prediction of the vortex state in superconductors was one of the main successes of Ginzburg-Landau theory
with dynamical U(1) gauge field. The vortex in the gauged Ginzburg-Landau theory
is nothing but the Z2 flux line in the Z2 gauge theory. The Ginzburg-Landau theory
without the dynamical U(1) gauge field fail to describe the real superconductors with dynamical electromagnetic interaction.
).
) 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. The ends of strings are defects which correspond to another type of excitations. Those excitations are the gauge charges and can carry Fermi
or fractional statistics.
The condensations of other extended objects such as "membranes
", "brane-nets" , and fractals also lead to topologically ordered phases and "quantum glassiness"
) theory may be the mathematical foundation of topological order. Quantum operator algebra
is a very important mathematical tool in studying topological orders. Some also suggest that topological order is mathematically described by extended quantum symmetry.
rotation symmetry can be used as the media of digital information storage. A hard drive made of ferromagnetic materials can store gigabyte
s of information. Liquid crystal
s that break the rotational symmetry of molecules find wide application in display technology; nowadays one can hardly find a household without a liquid crystal display somewhere in it. Crystals that break translation symmetry lead to well defined electronic bands
which in turn allow us to make semiconducting
devices such as transistor
s.
Different types of Topologically 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 in a technique known as topological quantum computing. A topologically ordered state is a state with complicated non-local quantum entanglement
. 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. 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 computations. Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made fault tolerant.
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. For example, the topological insulator
is a simple example of topological order. The boundary states of topological insulator
play a key role in the detection and the application of topological insulators.
theory is a cornerstone of condensed matter physics
. 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. Some suggest that topological order (or more precisely, string-net condensation) and the local bosonic (spin) models have the potential to provide a unified origin for photons, electrons and other elementary particles in our universe.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, topological order is a new kind of order (a new kind of organization of particles) in a quantum state that is beyond the Landau symmetry-breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
description. It cannot be described by local order parameters
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
and long range correlations. However, topological orders can be described by a new set of quantum number
Quantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
s, such as ground state degeneracy, non-Abelian Berry phases of degenerate ground states, quasiparticle
Quasiparticle
In physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
fractional statistics, edge states, topological entanglement entropy, etc. Strictly speaking, topological order is a pattern of long-range quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
in quantum states, defined via local unitary transformations. States with different topological orders cannot change into each other without a phase transition.
Background
Although all matter is formed by atomAtom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s, matter can have very different properties and appear in very different forms, such as solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
, liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
, superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
, magnet
Magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
, etc. According to condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
and the principle of emergence, the different properties of materials originate 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
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
in the materials.
Atoms can organize in many ways which lead to many different orders and many different types of materials. Landau symmetry-breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
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
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
), what happens is that the symmetry of the organization of the atoms changes.
For example, atoms have a random distribution in a liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
, so a liquid remains the same as we displace it 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
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
. In a crystal, atoms organize into a regular array (a lattice
Crystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
). A lattice remains unchanged only when we displace it by a particular distance (integer times of lattice constant
Lattice constant
The lattice constant [or lattice parameter] refers to the constant distance between unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c. However, in the special case of cubic crystal structures, all of the constants are...
), 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 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
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
theory is a very successful theory. For a long time, physicists believed that Landau symmetry-breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
theory describes all possible orders in materials, and all possible (continuous) phase transitions.
The discovery and characterization of topological order
However, since 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 people introduced chiral spin state. 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 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 . The proposed, new kind of order was named "topological order" . (The name "topological order" is motivated by the low energy effective theoryEffective theory
-Effective Theory:An Effective Theory is a theory based on reasonable assumption, if not actual evidence, for the purpose of moving on in an area of study. An effective theory provides a framework to explain certain effects, without the claim that the theory correctly models the underlying ...
of the chiral spin states which is a topological quantum field theory
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
(TQFT) ). New quantum numbers, such as ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
degeneracy
Degenerate energy level
In physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian for the system having more...
and the non-Abelian
Non-abelian
In theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group G, the elements of which do not obey the commutative law when they are multiplied. The original choice of G in the physics of electromagnetism was U, which is commutative.For a...
Berry's phase of degenerate ground states . were introduced to characterize the different topological orders in chiral spin states. Recently, it was shown that topological orders can also be characterized by topological entropy
Topological entropy (in physics)
The topological entanglement entropy, usually denoted by γ, is a number characterizing many-body states that possess topological order.The short form topological entropy is often used, although the same name in ergodic theory refers to an unrelated mathematical concept .A non-zero topological...
.
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 beyond 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.
Fractional quantum Hall (FQH) states were discovered in 1982 before the introduction of the concept of topological order. But FQH states are not the first experimentally discovered topologically ordered states. Superconductors discovered in 1911 are the first, which have a Z2 topological order (note that superconductivity can be described by the Ginzburg-Landau theory WITH dynamical U(1) gauge field, which is
a Z2 gauge theory --
an effective theory of Z2 topological order. The prediction of the vortex state in superconductors was one of the main successes of Ginzburg-Landau theory
with dynamical U(1) gauge field. The vortex in the gauged Ginzburg-Landau theory
is nothing but the Z2 flux line in the Z2 gauge theory. The Ginzburg-Landau theory
without the dynamical U(1) gauge field fail to describe the real superconductors with dynamical electromagnetic interaction.
).
Mechanism of topological order
A large class of topological orders is realized through a mechanism called string-net condensation . This class of topological orders can be classified by utilizing tensor category (or monoidal categoryMonoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
) 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. The ends of strings are defects which correspond to another type of excitations. Those excitations are the gauge charges and can carry Fermi
Fermi
Fermi may refer to*Enrico Fermi*Laura Fermi, Enrico Fermi's wife*Enrico Fermi Award*Fermi Paradox , an album by Tub RingTechnology:*Fermi, the codename for a CUDA architecture graphics card developed by Nvidia*Fermi Gamma-ray Space Telescope...
or fractional statistics.
The condensations of other extended objects such as "membranes
Membrane (M-Theory)
In theoretical physics, a membrane, brane, or p-brane is a spatially extended mathematical concept that appears in string theory and related theories...
", "brane-nets" , and fractals also lead to topologically ordered phases and "quantum glassiness"
Mathematical foundation of topological order
We know that group theory is the mathematical foundation of symmetry breaking orders. What is the mathematical foundation of topological order? The string-net condensation suggests that tensor category (or monoidal categoryMonoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
) theory may be the mathematical foundation of topological order. Quantum operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
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, Ferromagnetic materials that break spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
rotation symmetry can be used as the media of digital information storage. A hard drive made of ferromagnetic materials can store gigabyte
Gigabyte
The gigabyte is a multiple of the unit byte for digital information storage. The prefix giga means 109 in the International System of Units , therefore 1 gigabyte is...
s of information. Liquid crystal
Liquid crystal
Liquid crystals are a state of matter that have properties between those of a conventional liquid and those of a solid crystal. For instance, an LC may flow like a liquid, but its molecules may be oriented in a crystal-like way. There are many different types of LC phases, which can be...
s that break the rotational symmetry of molecules find wide application in display technology; nowadays one can hardly find a household without a liquid crystal display somewhere in it. Crystals that break translation symmetry lead to well defined electronic bands
Electronic band structure
In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
which in turn allow us to make semiconducting
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
devices such as transistor
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...
s.
Different types of Topologically 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 in a technique known as topological quantum computing. A topologically ordered state is a state with complicated non-local quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
. 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. 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 computations. Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made fault tolerant.
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. For example, the topological insulator
Topological insulator
A topological insulator is a material that behaves as an insulator in its interior or bulk while permitting the movement of charges on its surface....
is a simple example of topological order. The boundary states of topological insulator
Topological insulator
A topological insulator is a material that behaves as an insulator in its interior or bulk while permitting the movement of charges on its surface....
play a key role in the detection and the application of topological insulators.
Potential impact
Landau symmetry-breakingSpontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
theory is a cornerstone of condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
. 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
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
theory has so far indicated. Some suggest that topological order (or more precisely, string-net condensation) and the local bosonic (spin) models have the potential to provide a unified origin for photons, electrons and other elementary particles in our universe.
See also
- AKLT ModelAKLT ModelThe AKLT model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Affleck, Lieb, Kennedy and Tasaki provided crucial insight into the physics of the spin 1 Heisenberg chain...
- FractionalizationFractionalizationIn physics, fractionalization is the phenomenon whereby the quasiparticles of a system cannot be constructed as combinations of its elementary constituents...
- Herbertsmithite
- Implicate Order
- Quantum phase transitions
- Quantum critical pointQuantum Critical PointA quantum critical point is a special class of continuous phase transition that takes place at the absolute zero of temperature, typically in a material where the phase transition temperature has been driven to zero by the application of a pressure, field or through doping.Conventional phase...
- Quantum topologyQuantum topologyQuantum 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 associated with topological spaces and the related embedding of...
- String-net liquidString-net liquidIn condensed matter physics, a string-net is an extended object whose collective behavior has been proposed as a physical explanation for topological order by Michael A. Levin and Xiao-Gang Wen. A particular string-net model may involve only closed loops; or networks of oriented, labeled strings...
- String-net
- Topological defectTopological defectIn mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
- Topological entropy in physics
- Topological quantum field theoryTopological quantum field theoryA topological quantum field theory is a quantum field theory which computes topological invariants....
- Topological quantum numberTopological quantum numberIn physics, a topological quantum number is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations...
- Topological string theoryTopological string theoryIn theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry...
Fractional quantum Hall states
- D. C. Tsui and H. L. Stormer and A. C. Gossard, Phys. Rev. Lett., 48, 1559 (1982), "Two-Dimensional Magnetotransport in the Extreme Quantum Limit"
- R. B. Laughlin, Phys. Rev. Lett., 50, 1395 (1983), "Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations"
Chiral spin states
- V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett., 59, 2095 (1987), "Equivalence of the resonating-valence-bond and fractional quantum Hall states"
- Xiao-Gang Wen, F. Wilczek and A. Zee, Phys. Rev., B39, 11413 (1989), "Chiral Spin States and Superconductivity"
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, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces"
- Xiao-Gang Wen, Int. J. Mod. Phys., B4, 239 (1990), "Topological Orders in Rigid States"
- Xiao-Gang Wen, 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"
- Xiao-Gang Wen and Qian Niu, Phys. Rev. B41, 9377 (1990), "Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces"
- Xiao-Gang Wen, Phys. Rev. B, 43, 11025 (1991), "Gapless Boundary Excitations in the FQH States and in the Chiral Spin States"
- Alexei KitaevAlexei KitaevAlexei Kitaev is a professor of physics and computer science at the California Institute of Technology. He is best known for introducing the quantum phase estimation algorithm and the concept of the topological quantum computer while working at the Landau Institute for Theoretical Physics. For...
and John PreskillJohn PreskillJohn Phillip Preskill is an American theoretical physicist and the Richard P. Feynman Professor of Theoretical Physics at the California Institute of Technology ....
, Phys. Rev. Lett. 96, 110404 (2006), "Topological Entanglement Entropy" - Michael Levin and Xiao-Gang Wen, Phys. Rev. Lett. 96, 110405 (2006), "Detecting Topological Order in a Ground State Wave Function"
Effective theory of topological order
- Quantum field theory and the Jones polynomial, E. Witten, Comm. Math. Phys., 121, 351 (1989)
Mechanism of topological order
- Michael Levin, Xiao-Gang Wen, Phys. Rev. B, 71, 045110 (2005), String-net condensation: A physical mechanism for topological phases,
- Chamon, C., Phys. Rev. Lett. 94, 040402 (2005), Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection
- Alioscia Hamma, Paolo Zanardi, Xiao-Gang Wen, Phys.Rev. B72 035307 (2005), String and Membrane condensation on 3D lattices
- H. Bombin, M.A. Martin-Delgado, cond-mat/0607736, Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
Quantum computing
- Chetan Nayak, Steven H. SimonSteven H. SimonSteven H. Simon is an American theoretical physics professor at Oxford University and tutorial fellow of Somerville College, Oxford. Between 2000 and 2009 he was a director at Bell Laboratories. He is known for his work on Topological Phases of Matter, Topological Quantum Computing, and...
, Ady SternAdy SternAdy Stern is an Israeli physicist and an Associate Professor at the Weizmann Institute of Science. He is best known for his research relating to the theoretical aspects of the fractional quantum Hall effect...
, Michael FreedmanMichael FreedmanMichael Hartley Freedman is a 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 Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
, Sankar Das SarmaSankar Das SarmaSankar Das Sarma is an Indian-born American theoretical condensed matter physicist, who has worked in the areas of strongly correlated materials, graphene, semiconductor physics, low-dimensional systems, topological matter, quantum Hall effect, nanoscience, spintronics, collective properties of...
, http://www.arxiv.org/abs/0707.1889, 2007, "Non-Abelian Anyons and Topological Quantum Computation", Rev. Mod. Phys. 80, 1083 (2008). - A. Yu. Kitaev, Ann. Phys. (N.Y.), 303, 1 (2003), Fault-tolerant quantum computation by anyons
- Michael H. Freedman, Alexei KitaevAlexei KitaevAlexei Kitaev is a professor of physics and computer science at the California Institute of Technology. He is best known for introducing the quantum phase estimation algorithm and the concept of the topological quantum computer while working at the Landau Institute for Theoretical Physics. For...
, Michael J. Larsen, and Zhenghan Wang, Bull. Amer. Math. Soc., 40, 31 (2003), "Topological quantum computation" - Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, J. Math. Phys., 43, 4452 (2002), Topological quantum memory
- Ady SternAdy SternAdy Stern is an Israeli physicist and an Associate Professor at the Weizmann Institute of Science. He is best known for his research relating to the theoretical aspects of the fractional quantum Hall effect...
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, Phys. Rev. D68, 024501 (2003), Quantum order from string-net condensations and origin of light and massless fermions
- M. Levin and Xiao-Gang Wen, Fermions, strings, and gauge fields in lattice spin models., Phys. Rev. B 67, 245316, (2003).
- M. Levin and Xiao-Gang Wen, Colloquium: Photons and electrons as emergent phenomena, Rev. Mod. Phys. 77, Nu 12:19, 9 April 2009 (UTC)871 (2005), 4 pages; also, Quantum ether: Photons and electrons from a rotor model., arXiv:hep-th/0507118,2007.
- Zheng-Cheng Gu and Xiao-Gang Wen, gr-qc/0606100, A lattice bosonic model as a quantum theory of gravity,
Quantum operator algebra
- Yetter D.N., TQFTs from homotopy 2-types, J. Knot Theory 2 (1993), 113—123.
- 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 et al. A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity., Axiomathes, Volume 17, Numbers 3-4 / December, (2007), pages 353—408.,Springer,Netherlands,ISSN 1122-1151 (Print) 1572-8390 (Online). doi:10.1007/s10516-007-9012-1 .
- Ronald Brown, Higgins, P. J. and R. Sivera,:(2009), Nonabelian Algebraic Topology., vols.1 and 2, Ch.U. Press, in press.
- A Bibliography for Categories and Algebraic Topology Applications in Theoretical Physics
- Quantum Algebraic Topology (QAT)