Continuum model for solitons in polyacetylene

Abstract

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 $\ensuremath{\Delta}(x)={\ensuremath{\Delta}}_{0}tanh(\frac{x}{\ensuremath{\xi}})$ with $\ensuremath{\xi}$ as a variational parameter. It is shown within the weak-coupling limit that the soliton (creation) energy takes the minimum value $(\frac{2}{\ensuremath{\pi}}){\ensuremath{\Delta}}_{0}$ with $\ensuremath{\xi}=\frac{\ensuremath{\hbar}{v}_{F}}{{\ensuremath{\Delta}}_{0}}$, where $2{\ensuremath{\Delta}}_{0}$ 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 $\ensuremath{\Delta}(x)$ is an exact solution of the self-consistent Bogoliubov-de Gennes equation.