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I've been wondering whether there has been any experimental evidence for the existence of anyons or are they just objects of purely mathematical interest?

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    $\begingroup$ As far as I know, there has been no news. Some searches are on, but they are mostly test-beds (i.e. they are trying out the technology on the cheap and don't have the sensitivity to expect to see anything). $\endgroup$ Commented May 25, 2014 at 3:55
  • $\begingroup$ @dmckee That sounds like the beginning of an interesting answer, if some specialist in the field doesn't bite. $\endgroup$ Commented May 25, 2014 at 7:37
  • $\begingroup$ This question has been asked and answered here. This covers both the Abelian and non-Abelian cases. To my knowledge, nothing has changed since then. $\endgroup$ Commented May 26, 2014 at 2:49
  • $\begingroup$ It has been three years since this question was asked. Are there any recent developments in experiments that confirmed or ruled out the existence of anyons? The Wikipedia page on anyons, says, "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." $\endgroup$
    – SRS
    Commented Jul 2, 2017 at 12:42

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The only direct experimental evidence for anyons which the scientific community broadly accepts is the evidence described here for $\nu = 1/3$ Laughlin states in the fractional quantum Hall effect.

Experimental evidence for nonabelian anyons (which are necessary for universal topological quantum computing) is much weaker. The only promising lead seems to be the possible observation of Majorana zero modes trapped at the ends of superconducting wires, as described by the theoretical model of the Kitaev chain. There are two broad classes of systems that may have been found to support these: (a) indium antimonide wires studied at the University of Delft and subsequently duplicated at other institutions, and (b) ferromagnetic iron-atom chains studied at Princeton University.

(If I recall correctly, a year or two ago the Delft group improved their experiment to bring the zero-bias conductivity peak measurement - the key finding in support of Majorana zero modes - exactly in line with the theoretical prediction, but I don't have a citation for that.)

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