The velocity of sound of the SSH model
The Su-Schrieffer-Heeger (SSH) model is a simple tight-binding model
for polyacetylene. In this model, the coordinates
and motion of the atomic nuclei is treated classically, but
the 
-electron dynamics is treated fully quantummechanically in a
tight-binding approximation. The crucial
aspect of the model is that the coupling between the nuclear
coordinates is taken into account via the distance-dependent
hopping terms. The idea behind this is simply that the overlap
integrals between neighboring atomic orbitals increase when
the distance between neighboring atoms decreases.
As is well known, the important insight offered by the SSH model is that it allows for soliton solutions that separate the two degenerate ground states associated with the two ways in which the dimerization can take place (bonds S-L-S-L- and L-S-L-S-, where S stands for short and L for Long). There are spinless fractionally charged solitons as well as chargeless spin solitons. For an extensive overview and comparison with experimental data, we refer to the review of Heeger et al. cited below.
As discussed in this same review paper, several workers have analyzed the velocity of sound of the SSH model as a function of the coupling parameter, and concluded that the velocity of sound decreases linear in this coupling parameter from the zero coupling value. When Daniel Aalberts (then a postdoc in Leiden, now an assistant professor at Williams College), Ferry Vos (then a graduate student in Leiden, now at CMG) and I started working on the fast isomerization of rhodopsin, we found as a by-product an elegant method to evaluate the velocity of sound exactly: we just calculate the elastic constant of the SSH model by uniformly stretching it. From the usual formula for the velocity of sound in terms of the elastic constants, this gives us directly the velocity of sound! Our formula shows that the sound velocity renormalization due to the electron-phonon coupling in the model is exponentially small for weak coupling, contrary to claims which up to then had been made in the literature!
References
A. J. Heeger, S. Kivelson, J. R. Schrieffer and W.-P. Su, Solitons in Conducting Polymers, Rev. Mod. Phys. 60 781 (1988).
F. L. J. Vos, D. P. Aalberts and W. van Saarloos, A simple method for calculating the speed of sound in tight-binding models: application to the SSH model, Phys. Rev. B 53 R5986-R5989 (1996).
F. L. J. Vos, D. P. Aalberts and W. van Saarloos, The Su-Schrieffer-Heeger model applied to chains of finite length, Phys. Rev. B 53, 14922
(1996).
July 13, 1999
[Correlated systems] [Wim van Saarloos] [Instituut-Lorentz]