What's a charge density wave? I'm reading chapter 2 of Condensed Matter Field Theory by Alexander Altland, Ben D. Simons, Section on Interacting fermions in one dimension.
From what I understood, they considered the system of electrons in one dimension with Coulomb interaction. We are interested in low energy excitation, so they consider the dynamics near $\pm k_F$.
After approximating the free Hamiltonian with a linear approximation and writing everything in form of a density type operator, they transform the problem into a bosonic Hamiltonian, i.e., a new Hamiltonian
$$H=v_\rho \sum_q |q|b^\dagger_qb_q$$
containing boson creation and annihilation operators. The Hamiltonian resembles the form of a phonon or photon type Hamiltonian with dispersion $\omega=v_\rho |q|$. You call these excitations charge density waves (CDW).

Now, the way some other references talk about charge density waves is quite different from this. Like in Condensed Matter Physics R. Shankar, here they considered the neighboring electron interaction and set a gap equation via the mean-field theory approach. Similar things are written on Wikipedia. Here they considered Peierls' distortion due to lattice vibrations, and a similar approach has been taken in this paper.
How is the description so different? Or is it me that does not get the explanation given in Altland's book?

For example, a limiting case is explained in the R. Shankar's book as follows.
You consider a tight-binding + a nearest-neighbor interaction type Hamiltonian. The neighboring interaction is set by $U_0$. In the limit, when $U_0=\infty$, you put the electron with spacing between them so that the band is now fully filled (which was half-filled in the limit $U_0=0$). The two states (odd place spacing state or even place spacing state) are said to be CDW.
 A: I think a better way to think about charge density wave is phenomenologically. Namely, it is a collectively ordered phase of the charge degrees of freedom that spontaneously breaks the translation symmetry of the underlying lattice.
This can arise due to very different microscopic physics.
The classic example is the Peierls situation where the instability is driven by Fermi surface nesting which causes an enhancement of the Lindhard susceptibility  at the nesting vector. In this case you can think of the situation as the system spends energy in the form of strain when it distorts, but it saves more energy by reconstructing the Fermi surface and opening a gap.
There is the analogous phenomenon of spin density wave (SDW).  For example in elemental chromium, the SDW seems to be well understood in terms of the nesting picture. There is also SDW in the unconventional Fe based superconductors which often is described as being driven by Fermi surface nesting. However, in that case local moment behavior is also important and the SDW may not be of a purely itinerant nature.
If you want examples of CDW that I don't think can be understood as nesting, I would point to the CDW in the cuprate superconductors or the stripe phases in quantum hall systems.
My point is, I don't think there is any reason to believe that all CDW's should be able to be described by the same microscopic Hamiltonian or microscopic physics. They only become similar at the phenomenological level.
