# Talk:Anyon

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The use of Spin(2,1) as notation here seems not to be consistent with that at spinor.

Charles Matthews 09:19, 29 Nov 2003 (UTC)

Spin(p,q) is really the double cover of SO(p,q), not its universal cover in general... Of course, for Spin(3,1) and Spin(3), it makes no difference. Phys 13:25, 2 Dec 2003 (UTC)

The grammar in the last line of "In physics" is so bad that it's not even clear what is meant. — Preceding unsigned comment added by 128.111.9.44 (talk) 02:24, 20 July 2011 (UTC)

## Attempts at self promotion ?

I qualified the Stonybrook experiment as being controversial (which it is... very!). And removed reference to anyons in quantum optics which are very tangential to the field, if relevant at all. — Preceding unsigned comment added by 194.3.129.221 (talk) 23:04, 29 October 2011 (UTC)

## Too technical

Can we get some kind of introduction for lay people? I'm a physics major and I still can't make head or tail of this. Where is this thing on the scale from real to hypothetical? —Keenan Pepper 05:53, 30 March 2006 (UTC)

I can say that anyon-related phenomena are observed in solid state physics, so they're real. You might say the article is too technical, but I would say that this article is just not very good, which might explain why it's difficult to understand. Here's how I think of it: the universal cover of a circle (which is the rotation group in 2d) is the whole real line, and representations of the real line are just real numbers. Compare with the rotation group in 3d, whose universal cover is compact (it's a 3 sphere). Compact groups have discrete representations, so spin is discrete. The universal cover is actually only a double cover, so it takes half-integer values. The really interesting bit is how the action of interchange of 2d particles is path-dependent; particle exchange is a representation of the braid group in 2d, rather than just the symmetric group as in higher d. I dunno, if those rambling sentences will be of any use to you. It would be nice to have a good article on this, I'll put it on my list, but the list has been getting long, so I can't promise anything soon. -lethe talk + 06:25, 30 March 2006 (UTC)
I don't understand this either. I think someone needs to take a couple of steps back at the start of the article to explain the context. 81.19.57.146 12:08, 10 May 2006 (UTC)
Agreed, this is very possibly the single most baffling little article I've ever read here. And fuller1's explanation above is equally jargon-filled nonsense to a layman like me. I'm sure it all makes excellent sense to some - I can jargonize on certain topics with the best of them - but clearly the only people who could grasp this article are those who already understand its subject. So, here's another vote for some kindly pedagogue to take a good whack at this article....Eaglizard 19:58, 2 May 2007 (UTC)
If anyone can spare some 30minutes for this ... just write a comparision with Fermi-Dirac/Bose-Einstein Statistics, with two concrete examples where one can see that anyonic statistics are not completely contradicting to the FD/BE case, but rather a generalization that is possible in 2 dimensions. Maybe some more reference for observable effects would be nice. I think this should suffice to make this understandable --138.246.7.141 (talk) 12:49, 27 February 2008 (UTC)
OK, the problem here is not really that the article is too technical, but that it is too mathematical. Specifically, one does not expect even physics majors like Keenan, much less the General Interested Lay Audience (GILA), to know enough topology or even group theory (part of abstract algebra) to make sense of the explanations offered, either in the article itself or equally here on the Talk page by Lethe. Even math majors may not have enough topology to understand this! It follows that Lethe's approach to explanation might succeed if someone could help the GILA imagine what these covers and representations are. yoyo (talk) 01:47, 24 June 2008 (UTC)

I rearranged the whole article and tried to equip it with a pedagocical example. Please give me some feedback if I could get rid of the technicality problem. MKlaput (talk) 13:43, 27 February 2008 (UTC)

Please see my comment above. yoyo (talk) 01:48, 24 June 2008 (UTC)
You can't expect to be able to understand everything, e.g. representation theory, without some background. The cleanup has been done, there is now an accessible explanation of how it related to boson and fermion statistics. So I'm removing the tag.

## Missing references

I added some missing references 138.246.7.141 (talk) 12:49, 27 February 2008 (UTC)

## Start Class

It is great to see how the article developed since I first contributed quite a while ago. Is it still appropriate to have it be "start class"? I think in relation to the importance of the article, it is quite developed. There is a historical explanation of the term "anyon", it's relevance for condensed matter physics is stressed, there is a comprehensible physics explanation of anyonic statistics, there is one part on the Mathematics and there is plenty of references. What else do we need? --MKlaput (talk) 17:50, 18 August 2010 (UTC)

Could it be given more deteiled explanation of why fundamental group determines number of possible statistics? — Preceding unsigned comment added by 77.46.217.223 (talk) 20:44, 17 August 2011 (UTC)

## non-abelian

I'm surprised to see after all this time that the only mention of non-abelian anyons is in the references. I'll try restructuring the article a bit to include them in the body. Teply (talk) 21:44, 9 April 2012 (UTC)

## Diagrams

### Initial diagrams

Anyon as a charge-flux composite, in the case of a charge q around an infinitely long solenoid confining a "magnetic flux tube" entirely within the solenoid. No B field exists outside of the solenoid in this instance. This is a canonical illustration of the Aharonov-Bohm effect.
Anyon worldline topology in 2 + 1 dimensional spacetime. Interchange of two anyons by a clockwise rotation is inequivalent to an anticlockwise rotation.
In 3 + 1 dimensional spacetime, particle interchanges by clockwise and anticlockwise rotations have equivalent topology, because the third spatial dimensional dimension allows one world line to bend around the other.
Winding number of a charge-flux composite, with positive orientation in the anticlockwise sense, negative in the opposite sense.

Peshkin, M; Tonomura, A (1989). The Aharonov–Bohm effect. Springer-Verlag. ISBN 3-540-51567-4.CS1 maint: Multiple names: authors list (link)

(I don't have the book right now so can't give page numbers, but it's a short, thin, concise one easy to look through, so it's not really that essential in this case). Thanks. M∧Ŝc2ħεИτlk 06:31, 9 May 2013 (UTC)

### Feedback

The bit on Aharonov–Bohm effect is a bit misleading here because it remains a 3-D problem. Anyons are relevant to exchanging particles anywhere within the plane, not just around some solenoid passing through the origin.
The spacetime diagram showing rotation in 2 spatial dimensions is better, but still a little confusing. A much more useful activity right now would be to fix the turd that is the current state of the exchange operator article. I remember someone give a talk that showed an animation of two particles in a 3-D volume get exchanged once, then twice, then topologically deformed back. A similar kind of animation would do wonders on that page. An animation (not a static spacetime diagram) showing these exchanges in the plane could be very helpful here. The exchange operator isn't as well defined in 1-D, though you may be able to show exchange in a network Teply (talk) 17:29, 9 May 2013 (UTC)
Thanks for feedback, including the pointer to exchange operator and the paper.
The AB effect is in 3d, but I thought the charge-flux composite was the "standard" (?) example of an anyon. The caption can be always be rephrased.
In time I'll try to create the animation(s). I can easily produce SWF animations but annoyingly Wikimedia Commons cannot allow SWF files, and I can't yet seem to convert them to animated GIF which would be the best format (without paying for extra software - an obvious refusal). M∧Ŝc2ħεИτlk 07:45, 10 May 2013 (UTC)
As for what is the "standard" example I can't really say as anyonic statistics is largely viewed as exotica. The composite fermion idea has been around the longest to explain the FQHE, and maybe that's what you had in mind? More recently there's been a lot of excitement over the Majorana fermion as another realization. See [1][2][3] for starters. I find it to be a bit more intuitive, but maybe that's just me.
Part of the trouble may just be that your first two pictures show all 3 axes. For anyons, you really need to confine everything to a plane. The other trouble is that they appear to be single-particle pictures. If you have one particle in isolation, then you aren't really dealing with particle exchange. You can speak of the particle's rotation properties (e.g. integer vs. half-integer spin) or, as you have drawn it in your second figure, the different possible end states given different possible trajectories. The statistics comes from forcing particle exchange to occur around the magnetic region. This is what is really lacking in these articles relating to the spin-statistics theorem, that spin is a single-particle property whereas statistics is a multi-particle property. Teply (talk) 19:03, 10 May 2013 (UTC)
Thanks for clarifying the issue with the charge-flux composites - although the charge is drawn in the xy plane, the third axis does make an incorrect suggestion that anyons could be in 3d. The FQHE was not actually the idea, but it does happen to coincide there. Similarly the number of particles should be at least two. Incidentally the FQHE is a many-particle effect.
I'll read up more on the Majorana fermion soon also.
When you say "The statistics comes from forcing particle exchange to occur around the magnetic region.", presumably this refers to where the B-field is zero while the A-field is non-zero? In the above pics outside and around the solenoid?
Without attempting to detract, this article says "spin-statistics connection" while linking to spin-statistics theorem, what does the "connection" term have to do with anything?
Thanks, M∧Ŝc2ħεИτlk 15:13, 11 May 2013 (UTC)
Yes, by magnetic region I mean inside the solenoid, where B is nonzero. Anyway, have a look at figure 2 in Camino et al. (see the refs on this page), which shows the setup of an actual interferometer, the magnetic region, and the tunneling junctions that straddle it. A cartoon version of that could be a useful diagram here.
Whoever used the word "connection" probably didn't intend any special, technical definition of the term. There is a connection (relationship) between spin and statistics, as specified by the theorem. I'll change the wording.
If you're interested, another bad article worth improving is exchange symmetry. The identical particles article isn't so bad, and has some of the necessary material at hand. Teply (talk) 19:36, 12 May 2013 (UTC)
Laughlin quasiparticle interferometer atomic force micrograph, based on Fig.2.A of Camino et al.[1]
Laughlin quasiparticle interferometer scanning electron micrograph, based on Fig.2.B of Camino et al.[2]
As promised through a bit of procrastination, here are the Laughlin quasiparticle micrographs in SVG, let me know of any problems/improvements. M∧Ŝc2ħεИτlk 21:32, 17 May 2013 (UTC)
Pretty good for the image based on Fig.2.B. Maybe add some dashed lines to show the tunneling junctions. Otherwise great. Teply (talk) 06:44, 18 May 2013 (UTC)
Aren't those just the blue dots? M∧Ŝc2ħεИτlk 17:37, 18 May 2013 (UTC)
Yes, the new version looks alright. Write a caption and add it to the article. You don't always have to hide in the talk pages. Teply (talk) 04:50, 22 May 2013 (UTC)

### Exchange and orientation entanglement topological analogies

Worldlines of two identical particles in 3 + 1 dimensional spacetime, spiraling around each other during rotational particle exchange (twice) in the spacelike region. The third dimension allows one worldline to be deformed around the other, returning the system to the initial state, meaning the double exchange is topologically equivalent no exchange.

Hi Teply, I can finally produce animated GIFs by the stopframe method in Serif Drawplus X4 (oddly I couldn't get it to work properly before uninstalling and reinstalling the program, maybe just a glitch...), here is a rushed trial animation (not to be used anywhere for anything!!):

So as soon as possible I'll get the animations for the particle exchange done and fix the above pics, but please be aware it'll take a few weeks (exams)... Thanks again for your edits, feedback, and clarification. M∧Ŝc2ħεИτlk 21:54, 12 May 2013 (UTC)

Nooo, your figure shows exactly the opposite of its caption! You have two particles confined to a plane in 2+1 dimensional spacetime, and you are forced to rip one of the particles out of the plane to untie their braided world lines! I know representing 4 dimensions on a 2 dimensional computer screen is tough, but still...
Try this. Instead of a plane, show a translucent cube moving along. Then at the end, very importantly, keep both endpoints (final particle positions) fixed. Next, move only the part of the world lines that are within the final cube, to show untying in the third spatial dimension (as opposed to the time dimension). When that is done, relax the untied world lines back to parallel lines. Teply (talk) 03:17, 14 May 2013 (UTC)
I know it's not even shown, never mind implicitly suggested, but in actual intention the diagram is in 3d space + 1d time with only a plane in the 3-space to show exchange in that plane. You could imagine an invisible box, sliced through by the blue plane. I'll repair to this effect later, as said, it's a trial just to get used to GIF and see how well it uploads. M∧Ŝc2ħεИτlk 17:16, 14 May 2013 (UTC)
Not sure if it helps or hurts your understanding, but you can also picture this like the binasuan. Let your torso be the origin. Point your arms outward, palms up, elbows down.
First, let's check that your elbow has rotation characteristics of spin=1/2. With your palm facing up at all times, (counterclockwise) rotate your right forearm under the armpit and back around for a total 2π. Your elbow should now be pointed up. Again with your palm facing up at all times, rotate your right forearm over your shoulder and back around for another 2π. Your elbow should now be pointed down as it was when you started.
Now let's see what happens when we exchange arms, so to speak. Rotate your right arm until it is across your chest, pointing left, and rotate your left arm over your head so that it points right. That's one arm exchange. Rotate your right arm again (you're allowed to step over it, just don't spill the wine!) so that it returns to the right, and rotate your left arm again so that it returns to the left. That's two arm exchanges. Both your elbows should now be pointing down. In other words, each elbow has picked up a minus sign. That's 2 minus signs. Double negative gives the identity.
Of course to see that the statistics are fermionic you have to see what happens from just one arm exchange, not two. Since a 2π rotation of an arm inverts the elbow (multiplies by -1), a rotation by π is effectively multiplying by i. So when you're in that weird position after one exchange when your right arm points left and your left arm points right, each arm picks up a factor of i for an overall factor of -1.
The spin=1 case is easier. Let your whole arms held stiff be the vectors, and rotate your entire body. Teply (talk) 06:31, 14 May 2013 (UTC)
Here's a few more suggestions if you're feeling ambitious. I noticed a couple other kind of weak articles orientation entanglement and plate trick that try to discuss the same stuff. In orientation entanglement, the pictures with the mug don't really help me. Plate trick is also light on content, but a few of the links at the bottom are actually decent as they show both spin 1/2 and fermionic statistics. If you'd like to try making public domain versions of those or even just link the existing ones more prominently in the appropriate articles, that would be great. The only drawback I see with them is that they don't really show how 2π rotation or one exchange operation gives you the minus sign. Instead they show that 4π rotation or two exchange operations give you the identity. I guess it's left to the viewer to infer that the square root of 1 is supposed to be -1. Of course for anyonic statistics, the point is that you don't have enough freedom to move those belts around.
Of course my opinion is (and I think you'd agree) that every last one of these articles dealing with spin would benefit from just switching to rotations in geometric algebra already. Suppose you've got some object x that you want to rotate using rotor R. If memory serves me, you can say x has integer spin=s if it transforms as ${\displaystyle R^{s}xR^{\dagger s}}$ and s is the smallest integer with that property. For your usual multivector, you of course get spin=1 except if x is entirely scalar+pseudoscalar, in which case R commutes with x and spin=0. You can also say spin=1/2 if x transforms as ${\displaystyle Rx}$. (Fancier rules for spin=3/2, addition rules, and so on.) The spin=1/2 case is conceptually easier this way. Be bold... Teply (talk) 21:51, 14 May 2013 (UTC)
I definitely agree that a section on the description of spin particles in terms of GA would be nice. Do you know/have the rule for arbitrary half-integer spin, and addition (composition?) rules? M∧Ŝc2ħεИτlk 23:47, 21 May 2013 (UTC)
Let's not get too off topic. I'll move that discussion to your talk page. Teply (talk) 04:50, 22 May 2013 (UTC)

I quickly redrew the animation. Although the world lines untie in the spacelike region, it may still suggest that the worldlines move into the timelike region. Is it OK? M∧Ŝc2ħεИτlk 16:52, 17 May 2013 (UTC)

An old file retrieved and uploaded to be redrawn much, much nicer, cleaner, and clearer.
Forgot to add about orientation entanglement: the vimeo Dirac belt trick and vimeo belt trick for a two-particle exchange is a bit much for me to reproduce exactly to the same graphical quality (I'll try something later though). As a possible alternative, I drew a while back this crinkly and horrible animation for the orientation entanglement of a spinor, and intended to add it to that article but it was never finished to happen...
It's based on the "book and belt" trick in Penrose's Road to Reality, corresponding to the orientation entanglement for a spin-1/2 particle (no clue for higher spins...). M∧Ŝc2ħεИτlk 17:22, 17 May 2013 (UTC)

### Animations

Anticlockwise rotation
Clockwise rotation
Exchange of two particles in 2 + 1 spacetime. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in 2d space).

Here's two more for the 2+1 spacetime case, showing the clockwise and anticlockwise senses. I'll add them and the other 3d one above if approved, to this article (anyon) and exchange operator. M∧Ŝc2ħεИτlk 07:11, 26 May 2013 (UTC)

Could you make the animation slower? It is REALLY difficult to see what is happening at such speed... I had to look at it like 10 times, just because the dynamics of it were so fast. — Preceding unsigned comment added by 143.107.229.249 (talk) 11:18, 4 December 2014 (UTC)
Hi IP, saw your comment earlier and tried looking for the original file but it seems to be lost. I'll create another slower one. Best, M∧Ŝc2ħεИτlk 05:52, 5 December 2014 (UTC)

References

1. ^ Camino, F.; Zhou, Wei; Goldman, V. (2005). "Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics". Physical Review B. 72 (7). arXiv:cond-mat/0502406. Bibcode:2005PhRvB..72g5342C. doi:10.1103/PhysRevB.72.075342.
2. ^ Camino, F.; Zhou, Wei; Goldman, V. (2005). "Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics". Physical Review B. 72 (7). arXiv:cond-mat/0502406. Bibcode:2005PhRvB..72g5342C. doi:10.1103/PhysRevB.72.075342.

## thermal distribution analogue to fermi-dirac or bose-einstein statistics? do anyons obey Pauli exclusion or not?

I was doing some editing lately on the M-B, F-D, B-E statistics articles and I wondered what is the thermal distribution for anyons? It seems to me that any non-bosonic, indistinguishable particles would have to show Pauli exclusion principle, based on symmetry. In that case, anyons would have to follow identically the Fermi-Dirac thermal statistics. Is this true or am I just thinking about anyons in the wrong way? --Nanite (talk) 07:30, 7 August 2013 (UTC)

You are right that for any θ ≠ 0 there will be no such state as ψ ⊗ ψ, but your further thinking about anyons took the wrong way. For fermions (θ = 180°), all 2n “multiparticle” states of an n-state “particle” are orthonormal. For θ ≠ 180° they are not (and I am unsure about how the anyonic Fock space for θ ≠ 0, 180° will look and how many dimensions may it have), and consequently there will be no Fermi–Dirac statistics. Incnis Mrsi (talk) 10:49, 7 August 2013 (UTC)
Thanks Incnis Mrsi, I was suspecting that might be the case. I have followup question: Is it possible to speak of a thermal distribution of anyons, at all? Right now this article is listed beside thermal distributions in the Template:Statistical mechanics infobox, under "particle statistics", but there are no statistics (probabilities) mentioned in the article. I suspect there are two meanings of "particle statistics" being conflated here: 1) The phase factor of exchange, which is a really fundamental property, and 2) The correct counting method used in statistical thermodynamics when considering multiple particles falling into the same mode/orbital. Obviously the two are related for fermions/bosons (the latter being a consequence of the former), and that's why people got to calling the exchange factor "statistics". But on the other hand there is nothing statistical (probabilistic) about the exchange factor, since it is a built-in part of the mechanics of pure states. So I'm not so sure that para/anyon/braid statistics really count as statistical mechanics.
I found a related issue on the fermion article, where someone had said "all fermions obey Fermi-Dirac statistics" which (as I understand) isn't true since F-D statistics only apply for non-interacting fermions in thermal equilibrium. Again I think they mixed up the "statistics" of exchange with statistics proper. Nanite (talk) 10:39, 15 February 2014 (UTC)
While I think about the anyonic Fock space, answer please another question: is the concept of particle statistics incompatible with interaction potential between identical particles? You may look at boson where I asserted that it is compatible. Because it is an off-topical question here, post follow-ups to talk:Particle statistics. Incnis Mrsi (talk) 12:51, 15 February 2014 (UTC)

## Particle or quasiparticle?

Wouldn't it be better if the first sentence started- "In physics, an anyon is a type of quasiparticle..."? Bhny (talk) 00:36, 24 June 2014 (UTC)

Since there were no objections I made the change. Bhny (talk) 17:18, 24 June 2014 (UTC)

## No explanation of abelian

There is no explanation of why "abelian anyons" are abelian. The lead says is that it is explained below but it isn't. The word abelian doesn't even appear in the Anyon#Abelian_anyons section.

## create Fractional statistics page

create Fractional statistics page — Preceding unsigned comment added by 2A02:587:4110:4100:3408:7924:A4A:F99B (talk) 19:32, 16 November 2016 (UTC)

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