Effects of lattice fluctuations on Peierls distortion in a one-dimensional Holstein model

The effects of quantum and thermal lattice fluctuations on the Peierls distortion are treated through an electron-phonon scattering function introduced in a unitary transformation, and the perturbation calculation is implemented by means of the Green’s function method. The lattice-staggered ordering parameter for finite temperature, $m_p\mathrm{}\left(T\right),$ and the Peierls transition temperature $T_p,$ as functions of the phonon frequency and the electron-lattice coupling, have been calculated. Our main results are (1) a large ratio of the zero-temperature gap to $T_p$ agrees well with the experimental values for many Peierls systems; (2) the normalized ordering parameter $m_p{\mathrm{}\left(T\right)/m}_p\mathrm{}\left(0\right)\mathrm{}$ as a function of the normalized temperature ${T/T}_p$ has been calculated, and although it deviates from the mean-field results, it agrees qualitatively with the experimental ones of $({\mathrm{TaSe}}_4{)}_2\mathrm{I}$ and ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3;$ (3) the thermal lattice fluctuations may play an important role in suppressing the Peierls transition temperature even if the phonon frequency goes to zero.