Gap-edge states, broken particle-hole symmetry, and kink-antikink asymmetry in bounded Peierls systems

The continuum model for a Peierls system with a half-filled band is solved for a finite chain, the electron wave functions being required to vanish at the ends of the chain; results different from those found using periodic or antiperiodic boundary conditions are obtained. For the homogeneously dimerized state, the single kink (or antikink) state, and the single polaron state, there can be an electronic state at either the upper or the lower edge of the gap, without a corresponding state at the other edge. In agreement with the numerical result of Su on the Su-Schrieffer-Heeger model, the ground state of a chain with equal, odd numbers of electrons and ions contains a kink (or an antikink); also, for an odd chain, the kink energy differs from the antikink energy, neither value being the 2${\Delta }_0$/π found in previous work on the continuum model. The energy required to introduce a kink and an antikink is, however, 4${\Delta }_0$/π, as in previous work. The origin of the midgap state obtained upon the introduction of a kink (or an antikink) is simply explained in terms of the gap-edge states.