Soliton lattice structure and mid-gap band in nearly commensurate charge-density-wave states. II. Self-similar band-structure and coupling-constant dependence

We numerically diagonalize the one-dimensional Fr\"ohlich Hamiltonian by taking into account all higher harmonics associated with the charge-density-wave (CDW) formation self-consistently within the mean-field approximation. The overall band structures for various CDW states corresponding to a wide variety of the band fillings are derived to yield a self-similar structure, clarifying the interrelation between the mid-gap band and conduction (or valence) band. A detailed study in the nearly commensurate CDW states with the commensurabilities 2, 3, and 4 is given by further extending our previous calculation. We investigate, in particular, the electron-phonon coupling constant dependences of the physical quantities, i.e., the position of the mid-gap band, higher harmonics, kink width, localization length of the mid-gap wave function, and the CDW formation energy.