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 numbers, such as ground state degeneracy, quasiparticle fractional statistics, edge states, topological entropy, etc. Roughly speaking, topological order is a non-local quantum entanglement in quantum states. States with different topological orders can change into each other only through a phase transition. A Superconductor demonstrating the Meissner Effect. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In physics, a phase transition, (or phase change) is the transformation of a thermodynamic system from one phase to another. ... In physics, long-range order characterizes physical systems in which remote portions of the same sample exhibit correlated behavior. ... A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. ... In physics, a quasiparticle refers to a particle-like entity arising in certain systems of interacting particles. ... In mathematics and physics, an anyon is a type of projective representation of a Lie group. ... Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ...

A large class of topological orders is realized through a mechanism called string-net condensation. This class of topological orders is described and classified by a beautiful mathematical theory — tensor category theory. One finds that string-net condensation can generate infinity many different types of topological orders, which may indicate that there are many different new types of materials remaining to be discovered. In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object . ...

Background

Although all matter is formed by atoms, 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. Properties Mass: || ≈ 1. ... In jewelry, a solid gold piece is the alternative to gold-filled or gold-plated jewelry. ... A liquid will assume the shape of its container. ... Superfluidity is a phase of matter characterised by the complete absence of viscosity. ... Magnetic field lines of a bar magnet shown by iron filings on paper A magnet is an object that has a magnetic field. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... In physics, a phase transition, (or phase change) is the transformation of a thermodynamic system from one phase to another. ...

Atoms can organize in many ways which lead to many different orders and many different types of materials. With so many different orders, we need a general understanding of the orders. Landau symmetry-breaking theory provides such a general understanding. 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. Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... In physics, a phase transition, (or phase change) is the transformation of a thermodynamic system from one phase to another. ...

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, so a crystal has only discrete translation symmetry. The phase transition between a liquid and a crystal is a transition that changes 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 atoms have different symmetries in the two phases. A liquid will assume the shape of its container. ... It has been suggested that crystallization processes be merged into this article or section. ... The ordinary meaning of lattice is the basis for several technical usages A cherry lattice pastry A mathematical lattice that is a type of partially ordered set. ...

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. Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ...

However, in last twenty years, it has become more and more apparent that Landau symmetry-breaking theory may not describe all possible orders. In 1989, physicists introduced chiral spin state in an attempt to explain high temperature superconductivity. At first people 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. 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 symmetry description. This new kind of order was named topological order. A new quantum number, ground state degeneracy, was introduced to characterize the different topological orders in chiral spin states. (The name "topological order" is motivated by the low energy effective theory of the chiral spin states, which is a topological quantum field theory.) To meet Wikipedias quality standards, this article or section may require cleanup. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... The word degeneracy has more than one meaning: In general, degeneracy means reverting to an earlier, simpler, state In mathematics, a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ... In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). ... A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. ...

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. The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional systems of electrons subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ takes on the quantized values where e is the elementary charge and h is Plancks constant. ...

Applications

The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example, Ferromagnetic materials that break spin rotation symmetry can be used as the media of digital information storage. A hard drive made by ferromagnetic materials can store gigabytes of information on it. Liquid crystals 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 transistors. Topologically ordered states are a new class of materials may be even richer than symmetry breaking states. This may suggest an exciting potential for applications. A ferromagnet is a piece of ferromagnetic material, in which the microscopic magnetized regions, called domains, have been aligned by an external magnetic field (e. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... A gigabyte (derived from the SI prefix giga-) is a unit of information or computer storage equal to one billion bytes. ... Bold text Schlieren texture of Liquid Crystal nematic phase Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional liquid, and those of a solid crystal. ... In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ... In solid state physics, the electronic band structure (or simply band structure) of a solid is the series of forbidden and allowed energy bands that it contains. ... A semiconductor is a material with an electrical conductivity that is intermediate between that of an insulator and a conductor. ... Assorted component transistors The transistor is a solid state semiconductor device which can be used for amplification, switching, voltage stabilization, signal modulation and many other functions. ...

One theorized application would be to use topologically ordered states as media for 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 appartus 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. Molecule of alanine used in NMR implementation of error correction. ... Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ... Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ... Molecule of alanine used in NMR implementation of error correction. ... Fault-tolerance or graceful degradation is the property of a system that continues operating properly in the event of failure of some of its parts. ...

Potential impact

Why is topological order important? Landau symmetry-breaking theory is a cornerstone of condensed matter physics. It 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 a potential for topological order (or more precisely, string-net condensation) to provide a unified origin for photons, electrons and other elementary particles in our universe. Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... Properties The electron (also called negatron, commonly represented as e−) is a subatomic particle. ... In particle physics, an elementary particle is a particle of which other, larger particles are composed. ...

References

Fractional quantum Hall states:

• Two-Dimensional Magnetotransport in the Extreme Quantum Limit, D. C. Tsui and H. L. Stormer and A. C. Gossard, Phys. Rev. Lett., 48, 1559 (1982)
• Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, R. B. Laughlin, Phys. Rev. Lett., 50, 1395 (1983)

Chiral spin states:

• Equivalence of the resonating-valence-bond and fractional quantum Hall states, V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett., 59, 2095 (1987)
• Chiral Spin States and Superconductivity, Xiao-Gang Wen, F. Wilczek and A. Zee, Phys. Rev., B39, 11413 (1989)

Topological order:

• Quantum field theory and the Jones polynomial, E. Witten, Comm. Math. Phys., 121, 351 (1989)
• Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989)
• Topological Orders in Rigid States, Xiao-Gang Wen, Int. J. Mod. Phys., B4, 239 (1990)
• 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)
• Fractional Statistics and the Quantum Hall Effect, D. Arovas and J. R. Schrieffer and F. Wilczek, Phys. Rev. Lett., 53, 722 (1984)
• Gapless Boundary Excitations in the FQH States and in the Chiral Spin States, Xiao-Gang Wen, Phys. Rev. B, 43, 11025 (1991)

String-net condensation:

• Photons and electrons as emergent phenomena, Michael A. Levin, Xiao-Gang Wen, Rev. Mod. Phys., 77, 871 (2005)
• String-net condensation: A physical mechanism for topological phases, Michael Levin, Xiao-Gang Wen, Phys. Rev. B, 71, 045110 (2005)

Quantum computing:

• Fault-tolerant quantum computation by anyons, A. Yu. Kitaev Ann. Phys. (N.Y.), 303, 2 (2003)
• Topological quantum computation, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Bull. Amer. Math. Soc., 40, 31 (2003)
• Topological quantum memory, Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, J. Math. Phys., 43, 4452 (2002)