What are the key differences between a Wigner crystal and charge density wave? We know that Wigner crystal is a crystal formed due to interaction(kinetic energy is quenched). It is a crystal of electrons and therefore has periodical oscillations in the charge density. Therefore it seems it is legitimate to call it a charge density wave as well. 
However more often than not, the charge density wave(CDW) emerges as an instability from some free band electrons, when the interaction is added. In this picture, the electrons that constitute the CDW can move freely(when the CDW is formed, they subject to the CDW potential scattering.)
So the question is: what are the key differences between Wigner crystals and CDWs? When are they interchangeable, if possible? Or they are two extreme descriptions of the same thing, similar to the band theory case where a problem can be described by either the Bloch wavefunctions or the Wannier functions(it is just a matter of good starting point for a particular problem.) It seems to me the answer is yes. But I am not quite sure about it.
 A: Charge density waves and Wigner crystals are two ways of understanding the same broken symmetry. In the Wigner crystal picture we imagine the electrons sitting on lattice sites; the electronic charge density thus has broken translation and rotation symmetry. In the CDW picture we imagine an essentially uniform distribution of electrons that develops a slightly modulated charge density, $\rho(\mathbf{r})=\rho_0+\delta\rho\sum_i\exp(i\mathbf{q}_i\cdot\mathbf{r})$, which also has broken translation and rotation symmetry. In the Landau theory of phase transitions phases of matter are characterized by their broken symmetries (this paradigm fails in the case of topological phase transitions), so the Wigner crystal and CDW are actually the same phase of matter. It should be possible to tune from the CDW to the Wigner crystal and never encounter a phase transition. I could not find an example of an experiment which does this.
One caveat is that CDW is more generic than the Wigner crystal. Suppose we are thinking about a 2-dimensional system, such as a two-dimensional electron gas. Then the Wigner crystal is understood to break translation symmetry in both directions forming a triangular lattice for example. Similarly we can consider a CDW which breaks translation symmetry in both directions. However, we can also consider a CDW which breaks translation symmetry only along one direction of space, i.e., $\rho(\mathbf{r})=\rho_0+\delta\rho\exp(i\mathbf{q}\cdot\mathbf{r})$. Since this is a different broken symmetry are two different phases: one phase breaks translation symmetry only in one direction, e.g., the CDW with only one $\mathbf{q}_i$, and the other phase breaks translation symmetry in both directions, e.g., the CDW with multiple $\mathbf{q}_i$ which is the same phase as the Wigner crystal.
Quiz time: How many different Wigner crystal phases are there in 2D? How many CDW phases are there in 2D? Which of these phases are the same?
