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I heard that anyons can only be made in 2 dimensions because when you visualize the spacetime diagram of a 2-dimensional system with point particles, you can get braids, but if you do the same with a 3-dimensional system, you get only the un-braid since all braids can be unbraided in 4 dimensions.

But we can make braided surfaces in 4-dimensions by moving around rings in 3 dimensions! So does that mean we get 3 dimensional anyons if there are rings that behave like point-particle anyons?

I tried googling the keywords, but got no answers. I did find a video talking about braided surfaces using rings (Braids in Higher Dimensions - Numberphile).

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Indeed -- in theory, at least -- anyonic statistics does not so much require the ambient space to be 2-dimensional, as it requires the anyonic defects to have co-dimension 2 (hence dimension 2 less than that of the ambient space).

To give an impression, the following is a list of a few articles that consider 1-dimensional anyons (anyonic strings) in 3(+1) ambient dimensions (all in theory -- there is nothing yet on this in experiment/engineering, as far as I am aware, even though it is quite conceivable that anyonic string defects can exist in realistic materials):

  • J. Baez, D. Wise, A. Crans: "Exotic Statistics for Strings in 4d BF Theory", Adv. Theor. Math. Phys. 11 (2007) 707-749 (arXiv:gr-qc/0603085)

  • A. Bullivant, J. F. Martins, P. Martin: "Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory", Advances in Theoretical and Mathematical Physics 23 7 (2019) (arXiv:1807.09551)

  • L. Kong, Y. Tian, Z.-H Zhang: "Defects in the 3-dimensional toric code model form a braided fusion 2-category", J. High Energ. Phys. 2020 78 (2020) (arXiv:2009.06564)

In mathematical terms, passage from point-like anyons in 2(+1) dimensions to string-like anyons in 3(+1) dimensions corresponds to passing from the braid group (and its representations) to the "loop braid group" (pointer to an nLab page that I once wrote). This might also be called the "string braid group".

Going further up to any ambient dimension, a "$p$-brane" in $(p+2)(+1)$ dimensions -- often called a "defect brane" -- is expected to exhibit anyonic statistics, see for instance p. 65 of:

  • J. de Boer, M. Shigemori: "Exotic Branes in String Theory", Physics Reports 532 (2013) 65-118 (arXiv:1209.6056)

I am taking the liberty of closing by mentioning that just today we presented a proof of this expectation (assuming common hypotheses on the general mathematical rules of brane charges):

  • H. Sati, U. Schreiber: "Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential (TED) K-theory" (arXiv:2203.11838, nLab)
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In 3 dimensions, you can have both point particles and loops/rings/strings (all mean the same thing). There are several known braiding processes involving these two kinds of objects:

  1. A particle can move around a string.

  2. If there are two loops, one can "thread" one through the other to braid them. But this is actually not different from 1: the process can be smoothly deformed to the process 1 by first shrinking the loop to a point.

  3. More interestingly, imagine there are three loops, two of them are linked to the third. Then we can do braiding 2 to the the two loops. Because of the third loop, now the deformation from 2 to 1 is no longer possible: we can not shrink any of the linked loop to a point without hitting another one. So this is in some sense a new braiding for loops in 3D.

The three-loop braiding process has been studied quite extensively in the context of topological phases in 3+1. All of above are basically taken from the paper https://arxiv.org/abs/1403.7437.

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