# Why is the composite fermion not included in the anyon contents of FQH topological orders?

For example, both the $$\nu=1/3$$ Laughlin state and the Moore-Read state has a simple interpretation in terms of composite fermions, which are bound states of an electron and two fluxes.

Both the Laughlin states and the Moore-Read state also have anyons, since they are both topologically ordered. Laughlin states have Abelian $$ne/m$$ anyons, with $$m=1/\nu$$ and $$n, and Moore-Read state hosts non-Abelian anyons $$\sigma$$ with charge $$e/4$$ and a neutral fermion $$\chi$$.

However, composite fermions themselves do not appear in the anyon contents of either state, despite being such an important step in describing these states. My question is why.

Anyons are physical excitations in FQH states (by definition, something that can be trapped locally by a potential well), while composite fermions are just symbols in a theory of FQH states. Such symbols usually do not correspond to physical excitations. So they are not included as the physical excitations (the anyons) since they are not.

• Well, this answer is basically “they are not included because they are not”. My question is why the composite fermion does not correspond to physical excitations. In IQHE physical electrons form Landau levels and in FQHE composite fermions form Landau levels. Yet the former are in the spectrum of IQHE while the latter are not in the spectrum of FQHE. Are composite fermions confined, infinitely gapped, or something like that? Mar 17, 2019 at 0:24
• Composite fermions are not physical excitations, since composite fermions are always fermions while physical excitations are usually anyons. Mar 18, 2019 at 1:37
• But in general anyons can be fermions. Fermions do appear in the anyon content in many topological orders, such as Laughlin state, toric code, and Pfaffian state. Mar 18, 2019 at 3:07
• Fermion has Fermi statistics and anyon has fractional statistics. Fermion =/= anyon in general. In general anyons cannot be fermions. Mar 18, 2019 at 13:44
• No, this state is not in the physical Hilbert space, since it is not gauge invariant. If you apply composite fermion operator after gauge fixing, then the resulting state is some complicated superposition of the Anyons. Composite fermion picture can be misleading and fail to capture the essence of topological order. May 20, 2019 at 13:08

The question is: Where are composite fermions in the anyon content of FQHE? The answer is that composite fermions cannot be derived from the anyon content. Anyons, on the other hand, can be derived from composite fermions. In view of certain remarks made above: It is not debatable that composite fermions are physical particles. They have been observed in numerous experiments, as have many of their sates. In particular, the excitations of the principal n/(2pn+1) states have been established to be excited composite fermions. (E.g., the wave functions of excited composite fermions have an almost perfect overlap with the wave functions of excited states in exact diagonalization studies.) The anyon braid statistics arises (and can be derived straightforwardly within the composite fermion theory) when one seeks an effective theory of excitations by integrating out particles of the background fractional quantum Hall state.

• Thanks for the answer, Jain. I have no doubt that composite fermions are physically observable excitations in half-filled LL. I wonder if there is a more direct connection between anyons and composite fermions. Concretely, if one goes along Jain's (your) sequence approaching an compressible state, is there a connection between the gapped anyonic excitations in the FQH states in the sequence and the gapless CF excitations in the half-filled state? Oct 6, 2020 at 13:48

Just some comments on your reply to Jainendra. (I cannot directly comment under your reply because I am a new user.) As both Wen and Jain have mentioned, the anyons are the excitations of FQH states which have fractional braiding statistics and can be described by excited CF wave functions. On the other hand, if you attach a fractional number of quantized vortices to electrons, in principle, you can have some particles with anyonic statistics. We can write down such wave functions, in a framework consistent with CFs, as shown in our paper https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.115135 . We numerically confirm these intermediate states between two FQH states are gapped, as suggested by the adiabatic approach proposed by Greiter and Wilczek. But so far we treat these states as fictitious and we do not have a clear idea of their physical realizations.