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It is a well-established fact that topological excitations (anyons) in 2D topologically-ordered systems are described by unitary modular tensor categories, see, e.g., Appendix E in Kitaev (2006). One fundamental assumption behind this classification theory is that a specific anyon model (a topologically-ordered system) can only have a finite number of (topologically distinct) simple anyon types. But I wonder why must we assume this? Why can't there be a 2D system whose topological excitations are described by a braided fusion category with an infinite number of simple objects?

An infinite braided fusion category may seem crazy at first glance, but we have encountered it long before the concept of topological order was born: the category of finite dimensional representations of the Lie group $\rm SU(2)$ (or any finite dimensional semisimple Lie algebra) is an infinite symmetric fusion category (special case of infinite braided fusion category). Why can't they describe physical topological excitations?

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    $\begingroup$ well for starters you most certainly want finitely many vacua on closed manifolds, so... $\endgroup$
    – AccidentalFourierTransform
    Commented Aug 3 at 11:43

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In short, it's because anyon models are supposed to describe gapped topological phases, and a theory with infinitely many "anyon types" would necessarily be gapless.

This is perhaps easiest to see in the case of gauge theories. If $G$ is a finite group, then deconfined $G$ gauge theory defines a gapped topological phase, with $|G|^2$ anyon types if $G$ is Abelian. On the other hand, in $U(1)$ gauge theory, in the deconfined phase you have charges that are labelled by the irreducible representations of $U(1)$ (of which there are infinitely many), but deconfined $U(1)$ gauge theory necessarily has a gapless photon. One way to think about this is through a kind of "Goldstone's theorem" for spontaneously broken continuous higher-form symmetries:

https://arxiv.org/abs/1802.09512

Also, you can actually verify that if you construct a Hamiltonian on the lattice for $\mathbb{Z}_N$ gauge theory, then the ratio $\Delta/J$ goes to zero as $N \to \infty$, where $\Delta$ is the gap and $J$ is the "local strength" of the Hamiltonian (i.e. the sum of the norms of all terms that touch a given lattice site). This ratio is the appropriate dimensionless measure of being "close to gapless"

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  • $\begingroup$ Thanks for the nice answer! Two technical points confused me: (1). We know that for a finite group $G$, anyons in 2D deconfined $G$ gauge theory [or Kitaev's quantum double model based on $G$] are labeled by representations of the Drinfeld double $D(G)$. Is this still true for an infinite gauge group G? Then why are anyons in deconfined U(1) gauge theory labeled by objects of Rep[U(1)], not Rep[D(U(1))]? (2). Kitaev's $Z_{N}$-toric code is a sum of commuting projectors, so the spectral gap can only be an integer for any $N$, how can the gap go to zero as $N\to\infty$? $\endgroup$
    – Zhiyuan Wang
    Commented Aug 3 at 13:31
  • $\begingroup$ @Lagrange (2) It depends how you rescale the Hamiltonian as $N \to \infty$. I edited my answer to emphasize that the important point is that a certain dimensionless ratio goes to zero. $\endgroup$
    – Dominic Else
    Commented Aug 3 at 13:41
  • $\begingroup$ @Lagrenge (1) For a finite Abelian group $G$ the simple anyons are labelled by $G^* \times G$ -- where the first factor represents the irreducible representations and the second factor represents the fluxes. This indeed corresponds to $Rep[D(G)]$. If you apply this blindly to $G = U(1)$ you would get $\mathbb{Z} \times U(1)$. The thing is that because the second factor is continuous, it kind of get washed out by the gapless modes, and doesn't correspond to non-trivial superselection sectors any more. $\endgroup$
    – Dominic Else
    Commented Aug 3 at 13:47

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