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Order parameter of charge density wave in one dimension

  1. Aug 12, 2008 #1
    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
    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
    , it seems odd for me. Creating a right-moving electron and destroy a left-moving electron?
  2. jcsd
  3. Aug 16, 2008 #2
    It does indeed seem odd. Are you sure the hermitian conjugate isn't present?
  4. Aug 16, 2008 #3
    Yes, I copy the definition from J.Voits onedimensional fermi liquid p.1008

    I try to start from the density operator [tex]\rho(q)=\sum_k c_{k-q}^{\dagger}c_k[/tex].
    Then, decompose the Fermion operator[tex]c_k=\Theta(k)c_{R,k}+\Theta(-k)c_{L,k}[/tex].
    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[tex]c_{R,k-q}^{\dagger}c_{L,k}[/tex]seems to be the term [tex]\psi_R(x)^{\dagger}\psi_L(x)[/tex], but it is only for [tex]q\sim-2k_F[/tex].
    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 [tex]q\sim2k_F[/tex], 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 [tex]\rho_R(x)+\rho_L(x)[/tex] 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 [tex]0,\pm2k_F,\pm4k_F,...\pm2^nk_F[/tex] which means in one-dimensional fermion, the allowed periodicity for CDW is restrict to be some specific value?
    Last edited: Aug 16, 2008
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