Continuum model for solitons in polyacetylene

Solitons in a one-dimensional charge-density-wave system with half-filled electron bands are studied theoretically with a continuum model. This model is a continuum version of the one of polyacetylene recently considered by Su, Schrieffer, and Heeger (SSH). We have analyzed a variational solution with the displacement order parameter $\Delta \mathrm{}\left(x\right)={\Delta }_{0}tanh\left(\frac{x}{\xi }\right)$ with $\xi$ as a variational parameter. It is shown within the weak-coupling limit that the soliton (creation) energy takes the minimum value $\left(\frac{2}{\pi }\right){\Delta }_0$ with $\xi =\frac{\hslash v_F}{{\Delta }_0}$, where $2{\Delta }_0$ and $v_F$ are the dimerization energy gap and the Fermi velocity, respectively. These results agree quite well with numerical results by SSH for the discrete system. Furthermore, we show that the above $\Delta \mathrm{}\left(x\right)\mathrm{}$ is an exact solution of the self-consistent Bogoliubov-de Gennes equation.