Difference between charge density wave and charge distribution We can always see modulated charge density, the Friedel Oscillation, around an probe charge due to other electrons' response. Can this be called charge density wave (I believe not)? If not, what is the fundamental difference between this situation and charge density wave? Is Peierls Transition a precondition to charge density wave? If so, why? If not, what is the condition?
Thanks so much!
 A: Symmetry is the key to distinguish charge density wave (CDW) and other charge modulations.
CDW is not just a wavy pattern in the charge density as literally indicated by its name, it is an order that spontaneously breaks the translation symmetry. Note that the symmetry must be spontaneously broken but not be broken by hand. This means that the Hamiltonian of the system is translational symmetric, but the resulting ground state is not symmetric (due to the density modulation).
According to this criterion, Friedel oscillation is not a CDW state, because the Hamiltonian for Friedel oscillation includes a charge defect, and this defect breaks the translational symmetry explicitly on the Hamiltonian level, so the consequent charge modulation is not spontaneous, and therefore can not be called a CDW. Also the uneven charge distribution due to the multiple atoms in a unit-cell also does not qualify for CDW, because the atoms also breaks the translation symmetry on the Hamiltonian level explicitly, so the charge density modulation is not spontaneous.
Because the CDW state is by definition a spontaneous symmetry broken phenomenon, so it can not be smoothly connected to the high-temperature symmetric state, and a phase transition (like Peierls transition) is necessary before entering the CDW phase, as it is impossible to break the symmetry spontaneously without going through a phase transition. If a charge density modulation is not associated with such a transition, it will not be a CDW state.
A: You are right that modulated charge density like the Friedel oscillation cannot be called a charge density wave (CDW). In a CDW, the charge distribution does oscillate but with constant amplitude as opposed to a Friedel oscillation which decays out in space. The CDW is a stable phase usually arising from the opening of a gap, which is thermodynamically favourable even when compared to the cost of lattice distortion. While I would say that this is a generally occurring condition precursor to the CDW instability, I am not sure if there is a more overarching one.
Peierl's transition does set in a CDW instability in one dimensional (1D) systems. But because the transition is only for 1D systems and because there are genuinely 2D systems (not quasi 1D) like the cuprates with CDW phases, one might conclude that Peierl's transition is a sufficient but not necessary prerequisite.
