Infrared absorption of the charge-ordering phase: Lattice effects

The phase diagram of the half-filled spinless Holstein model for electrons interacting with quantum phonons is derived in three dimensions extending at finite temperature $T$ a variational approach introduced for the one-dimensional T=0 case. Employing the variational scheme, the spectral and optical properties of the system are evaluated in the different regimes that characterize the normal and ordered state. The effects of the charge-ordering ($CO$) induce a transfer of spectral weight from low to high energies in the conductivity spectra, as the temperature decreases or the strength of the electron-phonon ($el-ph$) interaction increases. The inclusion of effects of lattice fluctuations is able to smooth the inverse square-root singularity expected for the case of the mean-field approach and determines a subgap tail absorption. Moreover, in the weak to intermediate coupling regime, a two-component structure is obtained within the $CO$ phase at low frequency: the remnant Drude-like term and the incipient absorption band centered around the gap energy.