Order parameter of charge density wave in one dimension

[Yiping]

In one dimensional electron gas in charge density wave phase, as I know , the density of electrons will be periodic. The order parameter of charge density wave is written as
$$O_{CDW}(x)=\sum_s\psi_{L,s}^{\dagger}(x)\psi_{R,s}(x)$$
For Luttinger model, the \psi is the Fermion annihilation field operator.
I am not very familiar with Luttinger liquid, I try to understand why the order parameter of CDW would be written in this form. If I try to construct an order parameter for CDW by myself, I would guess the Fourier transform of electron density will be a feature for CDW, but I don't know the periodicity of the CDW. If the order parameter is correct, for CDW with any periodicity, I should find that <O_{CDW}>=1. I am not able to construct such order parameter by myself.
I can't understand why the order parameter will be the summation of
$$\psi_{L,s}^{\dagger}(x)\psi_{R,s}(x)$$
, it seems odd for me. Creating a right-moving electron and destroy a left-moving electron?

 

[Lonewolf]

It does indeed seem odd. Are you sure the hermitian conjugate isn't present?

 

[Yiping]

Yes, I copy the definition from J.Voits onedimensional fermi liquid p.1008

I try to start from the density operator $$\rho(q)=\sum_k c_{k-q}^{\dagger}c_k$$.
Then, decompose the Fermion operator$$c_k=\Theta(k)c_{R,k}+\Theta(-k)c_{L,k}$$.
Insert the relation into the density operator and expand it, I get four different kind of terms, with the same chirality or different chirality. After the Fourier transform, I think it is less odd for me.
The term$$c_{R,k-q}^{\dagger}c_{L,k}$$seems to be the term $$\psi_R(x)^{\dagger}\psi_L(x)$$, but it is only for $$q\sim-2k_F$$.
The CDW order parameter seems to include only this kind of term. I guess it is because the Hermitian conjugate only present the term with $$q\sim2k_F$$, which just shows the same periodicity of CDW in real space.

Except that, after I expand the density operator in this way, terms with the same chirality will be $$\rho_R(x)+\rho_L(x)$$ which will not give periodicity in real space, since the momentum is "almost" conserved. So the density modulation must comes from the cross term(momentum is not conserved<>translational invariance breaks.), which shows in the original question.
Then I am confused again, does that mean the momentum transfer, q, can only be $$0,\pm2k_F,\pm4k_F,...\pm2^nk_F$$ which means in one-dimensional fermion, the allowed periodicity for CDW is restrict to be some specific value?