Has the existence of anyons been experimentally verified? 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?
 A: 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.)
