Is it possible to make statements about bosonic/fermionic systems by taking the limit $\theta\to \pi$ or $\theta\to 0$, of an anyonic system?

Question

One might naïvely write the (anti-)commutation relations for bosonic/fermionic ladder operators as limits

$$ \delta_{k,\ell} = \bigl[ \hat{b}_{k}, \hat{b}_{\ell}^\dagger \bigr] = \hat{b}_{k} \hat{b}_{\ell}^\dagger - \hat{b}_{\ell}^\dagger \hat{b}_{k} = \lim_{\theta\to\pi} \Bigl( \hat{b}_{k} \hat{b}_{\ell}^\dagger + e^{i\theta}\cdot\hat{b}_{\ell}^\dagger \hat{b}_{k} \Bigr) $$ $$ \delta_{k,\ell} = \bigl\{ \hat{c}_{k}, \hat{c}_{\ell}^\dagger \bigr\} = \hat{c}_{k} \hat{c}_{\ell}^\dagger + \hat{c}_{\ell}^\dagger \hat{c}_{k} = \lim_{\theta\to 0} \Bigl( \hat{c}_{k} \hat{c}_{\ell}^\dagger + e^{i\theta}\cdot\hat{c}_{\ell}^\dagger \hat{c}_{k} \Bigr). $$ I.e. as limits of Abelian anyonic commutation relations. Assuming now that some system could be solved for anyons with $0 < \theta < \pi$, would taking the limits of e.g. the energy eigenstates for $\theta\to \pi$ yield in general the correct eigenstates of the bosonic system (which might be harder to solve directly)?

I'm inclined to think it would work, but after all, the whole Fock space looks different depending on $\theta$, with all kinds of possible topological nontrivialities.

Answer 1

There is no such thing as "Abelian anyonic commutation relations", in the sense that the "Abelian anyonic commutation relations" that you write down does not describe Abelian anyons. So the starting point of the question is not valid.

Also anyons do not have a Fock space description. The standard many-body text books stress on Fock space too much, which lead people to think about many-body systems only in terms of Fock space. Such Foack-space picture can only describe a very small subset of many-body states. Most many-body states (the interesting ones) require a new picture (such as tensor network) to visualize.