# 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.

• What is "them" in "spacing between them"? Electrons (plural)? Something else? May 12 at 18:33
• I meant, When you fill the state, you leave one form unfilled between the two electrons. May 12 at 18:59