A second well-studied example is the charge density wave (CDW)[12]. A CDW is essentially a sort of anisotropic electron crystal. They tend to occur in compounds composed of relatively weakly coupled chains. This was explained by a simple argument due to Peierls. He considered the energy of a one-dimensional electron system coupled to phonons. If the coupling were zero, the ground state of the system would be an undistorted chain and a filled Fermi sea. When the two densities are coupled, however, the total energy may actually be lowered by creating a periodic distortion of the electron density. To see this, let us suppose that the atoms in the chain develop a phonon distortion
where w is the phonon coordinate, 
is the Fermi
wavevector, and 
is an arbitrary phase. The electrons
experience a periodic potential from this distortion, which is crudely
modeled by the single-particle Hamiltonian
where g is some interaction constant. The single-particle
eigenstates are then Bloch waves, and the spectrum
develops gaps at the edges of the Brillouin zone.
Near the edge 
, the spectrum is
where the Fermi velocity 
and the gap 
. The electronic energy is thus lowered by
This lowering is logarithmically enhanced over the energy gain from the phonon distortion
where K is the one-dimensional elastic modulus of the chain. Because of the logarithmic enhancement of the electronic contribution due to the gap, the net energy is indeed always lowered by the distortion.
Physically, the system simply creates a periodic potential for the electrons, which then fall into that potential. Provided electron-electron interactions are not too strong, opposite spin electrons pair up in each minima of this potential (when repulsive electron-electron interactions dominate, quasi-one-dimensional systems instead form spin density waves, which have more complex properties). Since this distortion is tied to the electron density, we can, if we like, forget about the lattice distortion entirely and treat the CDW as simply a quasi-1d (paired) electron solid. The only residual effect of the lattice is to change the effective mass and damping of the local charge degrees of freedom of this solid.