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From the literature I've read, Fermi-nesting is a mechanism that can lead to charge density wave (CDW) formation in which a gap is created in the Fermi surface (which correspons with a nesting wavevector, $2k_F$) to compensate for the energy cost in the periodic lattice distortion. In the 1D case of a linear chain of atoms, the non-interacting electronic susceptibility (Lindhard function) diverges to infinity which corresponds to a softening of the frequency of a Kohn anomaly phonon at wavevector $2k_F$. I'm struggling to understand the reason of the softening. I've read that it has something to do with renormalisation of the phonon dispersion but I'm not sure what that means in physical context.

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For example, I thought the phonon softening occured due to a reduction in the density of states of electrons at the Fermi level which may have reduced the phonon-electron interactions and hence reduced the frequency of the phonons with wavevector $2k_F$ but I do not think that's correct.

Can someone explain why the softening of the Kohn anomaly phonon at $2k_F$ takes place (in regards ot CDW formation)? Is it that strong electron-phonon interactions (as implied by the divergence of the electronic susceptibility) means that the Fermi surface is rigidly shifted and hence the frequency is zero (in the most extreme case, like that of the 1D case)?

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