Klein factors are the operators which make sure that the anticommutation between the different species is correct during the bosonization procedure. According to this famous review by Jan Von Delft, they are the operators responsible for raising or lowering the fermionic number. But most of the literature often tend to ignore this factor. Recently I have come across this paper where they have given an explicit representation of the Klein factors (in appendix A). But unfortunately, this representation can't connect the different fermionic particle number states. I am really confused with what exactly is Klein factors are. Please help me resolve my doubt.
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Preface
Fully understanding Klein factors is one of those things that doesn't really matter, except for when it does... I think that's why you'll find many different, seemingly incompatible versions in the literature. I also found this confusing when I was learning bosonization, so I feel you. Eventually I stumbled on the lecture notes by Schultz, Cuniberti, and Pieri, which proved very helpful. (In the arXiv version, the relevant discussion is after Eq. (4.17), and in the appendix. In the version published by Springer, it's after Eq. (2.96), and in the appendix. Haldane's derivation in his 1979 paper (paywalled) might also be useful, although it's considered less straight-forward.)
What's the right definition?
We often start with a fermionic Hamiltonian, and bosonize that. As you say, the idea of Klein factors is to ensure that the bosonized version faithfully replicates the fermionic one. There are two things the bosonic fields cannot do, for which the cure is to introduce the Klein factors
- Ensuring anticommutation between different fermion species
- Connect different fermion number sectors
The definition given in von Delft and Schoeller achieves precisely this. When we are interested in thermodynamic properties (e.g. correlation functions as $L\rightarrow \infty$), however, the fact that the Klein factors change the particle number amounts to a vanishing shift in $k_F$. Hence, this structure is often neglected, and one is left with a much simpler-looking Klein factor. Often a representation in terms of Dirac gamma matrices is used, which obeys the familiar Clifford algebra. (These are often viewed as some kind of fictitious Majorana fermion.) In my experience this is the most common form of Klein factors encountered in the literature.
Suppose then that we work in the thermodynamic limit, and have opted for a "Majorana" representation for our Klein factors, which I'll denote $\eta$. We bosonize all the terms in our Hamiltonian, and decide to group them according to their Klein factor parts. If these (for a lack of a better word) "Klein factor prefactors" commute, we can assume we are in a certain sector of this $4\times 4$ Klein space, and replace these prefactors by their eigenvalues. This can be considered a sort of gauge fixing.
For a good example of how this works in practice, see the appendix of the Schultz et al. lecture notes. If we actually do this, replace these prefactors by their eigenvalues, why, we have achieved a purely bosonic Hamiltonian! A considerable part of the literature makes this jump right away, and assumes that is possible - ignoring the Klein factors, as it were. To be fair, this is usually possible, at least in many physically relevant cases, such as fermions with internal SU(N) symmetry.
So long story short, depending on what system you aim to describe, different appropriate choices of Klein factors may be available.
About the Klein factors in the Teo and Kane paper
I haven't checked this carefully, but I expect that they implicitly assuming they're in the thermodynamic limit. Then all we need the Klein factor for is to get the anticommutation relations right, which can be achieved in different ways (as is noted in e.g. Sénéchal's notes on bosonization). You'll note that the form they're using in Eq. (A3) is very similar to the way anticommutation relations are enforced in the Jordan-Wigner transformation from spins to fermions. The advantage of this kind of Klein factor* is that there is no need to introduce an extra Hilbert space for the Klein eigenstates to live in.
*I'm not sure that's an appropriate name, but they use it, so I'll stick with it.
Anyon
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