text: In 1954, Herbert Fröhlich proposed a microscopic theory, in which energy gaps at ±k would form below a transition temperature as a result of the interaction between the electrons and phonons of wavevector Q=2k.  Conduction at high temperatures is metallic in a quasi-1-D conductor, whose Fermi surface consists of fairly flat sheets perpendicular to the chain direction at ±k. The electrons near the Fermi surface couple strongly with the phonons of nesting wave number Q = 2k. The 2k mode thus becomes softened as a result of the electron-phonon interaction.  The 2k phonon mode frequency decreases with decreasing temperature, and finally goes to zero at the Peierls transition temperature. Since phonons are bosons, this mode becomes macroscopically occupied at lower temperatures, and is manifested by a static periodic lattice distortion.  At the same time, an electronic CDW forms, and the Peierls gap opens up at ±k. Below the Peierls transition temperature, a complete Peierls gap leads to thermally activated behavior in the conductivity due to normal uncondensed electrons.
  However, a CDW whose wavelength is incommensurate with the underlying atomic lattice, i.e., where the CDW wavelength is not an integer multiple of the lattice constant, would have no preferred position, or phase φ, in its charge modulation ρ + ρcos[2kx – φ]. Fröhlich thus proposed that the CDW could move and, moreover, that the Peierls gaps would be displaced in momentum space along with the entire Fermi sea, leading to an electric current proportional to dφ/dt. However, as discussed in subsequent sections, even an incommensurate CDW cannot move freely, but is pinned by impurities. Moreover, interaction with normal carriers leads to dissipative transport, unlike a superconductor.