# Collective modes of charge density wave

The question is about collective modes of charge density waves, i.e., amplitude and phase fluctuations $$\delta,\phi$$ of the order parameter $$\Delta(x,t)=(\Delta_0+\delta)e^{i\phi}$$.

I read on p.1 of this paper that amplitude mode and phason mode can be expressed using the phonon normal coordinate $$Q_{q\pm2k_F}$$ respectively as ($$2k_F$$ is the density wave vector) $$\frac{1}{\sqrt{2}}(Q_{q+2k_F}\pm Q_{q-2k_F}).$$ But how to obtain this result? It's not quite obvious to me.

I also found in another review paper (p. 1135, eq(3.1)) the following. To the lowest order in the fluctuations $$\delta,\phi$$, the amplitude mode corresponds to $$\Delta_{2k_F}+\Delta_{-2k_F}=2\Delta_0+2\delta$$ and the phason mode corresponds to $$\Delta_{2k_F}-\Delta_{-2k_F}=2\Delta_0\Delta\phi$$. But I don't find a clear definition of $$\Delta_{2k_F}$$ in the paper. Still it's not clear how this is obtained.