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(1) For anyons to be created locally in a physical model they must be created in groups such that the local excitation is a boson or a fermion. However, the local excitation can fractionalize into anyonic parts which can propagate independently. In terms of second quantized operators the expectation is that the the local fermionic/bosonic degree of freedom ...

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For the first part, although it does not quite make sense to ask "the probability to have a Majorana in position x" (there are no "Majorana" in the system; the system only has electrons), the wavefunction $|u|^2$ ($|v|^2$) does have a physical meaning as the weight for a single electron (hole) excitation, see the answers in a closely related question: Can ...

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Clearly the "TMI" and the slave-rotor mean-field state are very different, because the TMI, as you assume, has no topological degeneracy while the other state is topologically ordered. However, I feel this answer is not very meaningful without seeing more details of the slave-rotor mean-field state. I'm afraid this is not a very well-known (or even ...

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