Charge density fluctuations and the softening of phonons in electron–phonon systems

The theory of charge–density fluctuations and phonon-softening is formulated exactly using the diagrammatic method for simple electron–phonon systems without many band effects and Umklapp processes. These electron–phonon systems are classified to three cases according to the strength of the electron–electron interaction and the electron–phonon interaction. The criteria for occurrence of the charge-density wave state and periodic lattice distortion are discussed in each case. They are given using the partial summation of diagrams. Here the phase transition is assumed to be of second order.