Dragon curve

Template:Distinguish2 Template:Statistical mechanics In physics, an anyon is a type of particle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons; the operation of exchanging two identical particles may cause a global phase shift but cannot affect observables. Anyons are generally classified as abelian or non-abelian, as explained below. Abelian anyons have been detected and play a major role in the fractional quantum Hall effect. Non-abelian anyons have not been definitively detected although this is an active area of research.

Abelian anyons

In space of three or more dimensions, indistinguishable particles are either fermions or bosons, according to their statistical behaviour. Fermions obey the so-called Fermi–Dirac statistics while bosons obey the Bose–Einstein statistics. In the language of quantum mechanics this is formulated as the behavior of multiparticle states under the exchange of particles. This is in particular for a two-particle state (in Dirac notation):

$\left|\psi _{1}\psi _{2}\right\rangle =\pm \left|\psi _{2}\psi _{1}\right\rangle$

(where the first entry in Template:Braket is the state of particle 1 and the second entry is the state of particle 2. So for example the left hand side is read as "Particle 1 is in state ψ₁ and particle 2 in state ψ₂"). Here the "+" corresponds to both particles being bosons and the "−" to both particles being fermions (composite states of fermions and bosons are irrelevant since that would make them distinguishable).

In two-dimensional systems, however, quasiparticles can be observed which obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by Jon Magne Leinaas and Jan Myrheim of the University of Oslo in 1977. In our above example of two particles this looks as follows:

$\left|\psi _{1}\psi _{2}\right\rangle =e^{i\,\theta }\left|\psi _{2}\psi _{1}\right\rangle \,,$

with i the imaginary unit and θ a real number. Now |e^iθ| = 1, e^2πi = 1, and e^πi = −1. So in the case θ = π we recover the Fermi–Dirac statistics (minus sign) and in the case θ = 0 (or θ = 2π) the Bose–Einstein statistics (plus sign). In between we have something different. Frank Wilczek coined the term "anyon" to describe such particles, since they can have any phase when particles are interchanged.

We also may use θ = 2πs with particle spin quantum number s, with s being integer for bosons, half-integer for fermions, so that

$e^{i\,\theta }=e^{2\,i\pi s}=(-1)^{2\,s}$   or   $\left|\psi _{1}\psi _{2}\right\rangle =(-1)^{2\,s}\left|\psi _{2}\psi _{1}\right\rangle .$

At an edge, fractional quantum Hall effect anyons are confined to move in one space dimension. Mathematical models of one dimensional anyons provide a base of the commutation relations shown above.

Just as the fermion and boson wavefunctions in a three-dimensional space are just 1-dimensional representations of the permutation group (S_N of N indistinguishable particles), the anyonic wavefunctions in a two-dimensional space are just 1-dimensional representations of the braid group (B_N of N indistinguishable particles). Anyonic statistics must not be confused with parastatistics which describes statistics of particles whose wavefunctions are higher dimensional representations of the permutation group.

Experiment

Laughlin quasiparticle interferometer scanning electron micrograph of a semiconductor device. The four light grey regions are Au/Ti gates of undepleted electrons, the blue curves are the edge channels from the equipotentials of these undepleted electrons. The dark grey curves are etched trenches depleted of electrons, the blue dots are the tunneling junctions, the yellow dots are Ohmic contacts. The electrons in the device are confined to a 2d plane.

Daniel Tsui and Horst Störmer discovered the fractional quantum Hall effect in 1982. The mathematics developed by Leinaas and Myrheim proved to be useful to Bertrand Halperin at Harvard University in explaining aspects of it. Frank Wilczek, Dan Arovas, and Robert Schrieffer verified this statement in 1985 with an explicit calculation that predicted that particles existing in these systems are in fact anyons.

In 2005 a group of physicists at Stony Brook University constructed a quasiparticle interferometer, detecting the patterns caused by interference of anyons which were interpreted to suggest that anyons are real, rather than just a mathematical construct. However, these experiments remain controversial and are not fully accepted by the community.

With developments in semiconductor technology meaning that the deposition of thin two-dimensional layers is possible – for example in sheets of graphene – the long term potential to use the properties of anyons in electronics is being explored.

Non-abelian anyons

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Here is my homepage ... new launch ec In 1988, Jürg FröhlichTemplate:Verify source showed that it was possible in a valid particle theory for the particle exchange operation to be non-commutative (non-Abelian statistics). Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in fractional quantum Hall effect. While at first non-abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when Alexei Kitaev showed that non-abelian anyons could be used to construct a topological quantum computer. As of 2012, no experiment has conclusively demonstrated the existence of non-abelian anyons although promising hints are emerging in the study of the ν = 5/2 FQHE state.

Topological basis

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In more than two dimensions, the spin–statistics theorem states that any multiparticle state of indistinguishable particles has to obey either Bose–Einstein or Fermi–Dirac statistics. For any d > 2, the group SO(d,1) (which generalize the Lorentz group), and also Poincaré(d,1), have Z₂ as their first homotopy group. The cyclic group consisting of two elements, Z₂, therefore only two possibilities remain. (The details are more involved than that, but this is the crucial point.)

The situation changes in two dimensions. Here the first homotopy group of SO(2,1), and also Poincaré(2,1), is Z (infinite cyclic). This means that Spin(2,1) is not the universal cover: it is not simply connected. In detail, there are projective representations of the special orthogonal group SO(2,1) which do not arise from linear representations of SO(2,1), or of its double cover, the spin group Spin(2,1). These representationsTemplate:Clarify are called anyons.

This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group is SO(2) has an infinite first homotopy group.

This fact is also related to the braid groups well known in knot theory. The relation can be understood when one considers the fact that in two dimensions the group of permutations of two particles is no longer the symmetric group S₂ (with two elements) but rather the braid group B₂ (with an infinite number of elements). The essential point is that one braid can wind around the other one, an operation that can be performed infinitely often, and clockwise as well as counterclockwise.

A very different approach to the stability-decoherence problem in quantum computing is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.

See also

• Plekton
• Flux tube
• Anyonic Lie algebra

References

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Further reading

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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