Soliton lattice structure and midgap band in nearly commensurate charge-density-wave states

The incommensurate charge-density-wave (CDW) states for systems with a nearly-half-filled, a nearly-third-filled and a nearly-quarter-filled band are investigated theoretically by solving the Fröhlich model in one dimension within a mean-field approximation. The Hamiltonian matrices with sizes as large as 100×100 are numerically diagonalized in momentum space in a self-consistent manner, taking into account all higher harmonics. The stable CDW states are thus determined, yielding the energy-gap structure, the electron-density modulation, the order parameters with various higher harmonic components, and the degree of localization of the eigenfunctions. It is shown that the midgap band inside the main Peierls gap always appears in nearly commensurate CDW states. This is attributed to the soliton (or kink) lattice structure of the electron-density modulation. A universal form of the electron-density modulation is deduced in the commensurate limit. Peculiar peaks found in the recent absorption experiment on orthorhombic ${\mathrm{TaS}}_3$ are interpreted in terms of the midgap band. Possible experiments on other quasi-one-dimensional CDW materials such as ${\mathrm{K}}_{0.3}$ ${\mathrm{MoO}}_3$ are proposed.