# Optical Response of an Interacting Polaron Gas in Strongly Polar Crystals
## 5 Conclusions  
In the present work, we revisit the optical response of a polaron gas in complex polar crystals using the random phase approximation extended for an interacting electron-phonon system. This extension results in a modified many-polaron optical conductivity with an effective electron-electron interaction accounting for the dynamic screening by LO phonons. For a more realistic calculation relevant for comparison with experiment for strontium titanate, the phonon dielectric function contains several optical-phonon modes that have been identified in SrTiO\({}_{3}\). A distinctive low-frequency peak of the many-polaron optical conductivity in a polar medium appears when a crystal is highly polar, \(\varepsilon_{0}/\varepsilon_{\infty}\gg 1\), which is realized in strontium titanate. As can be seen from the obtained spectra of the optical conductivity, the dynamic screening leads to an appearance of this peak which is close to the experimental \"130-meV feature\", except for the highest available density. Moreover, its width and shape asymmetry are remarkably similar to those of the experimental peak, including even fine details such a small kink at the shoulder above the maximum. Also, the whole shape of the spectrum at least for the two lower densities is similar to the experimental results, containing both the low-frequency peak and the \"plasmon-phonon\" peak due to undamped plasmons. This similarity makes the dynamic screening mechanism for the low-frequency peak convincing. As can be seen from the comparison of the results [13] obtained without account of dynamic screening with those of the present work, we have achieved now much better agreement with the experiment remaining completely within a single physical mechanism exploited for the theoretical interpretation of the many-polaron optical response. Strontium titanate exhibits probably the strongest manifestation of the effect of dynamic screening with respect to other known polar crystals because of an extremely large \(\varepsilon_{0}/\varepsilon_{\infty}\) ratio, especially at low temperatures. There is also a remarkable agreement between the present theory and the experimental results [2] in what concerns the high-frequency dependence of the optical conductivity (in the range \(\hbar\Omega\sim 1\) eV), achieved without adjustment, using reliable material parameters known from literature. This agreement is in line with experimentally substantiated conclusion [39] that polarons in SrTiO\({}_{3}\) are large rather than small. The developed method has of course less predictive accuracy than ab initio calculations supported by an outstanding progress of the computation power. The semianalytic approaches have, as shown above, an advantage to give a transparent physical picture of observable features of the many-polaron response spectra. In this connection, it would be interesting to compare the obtained results with those calculated from the first principles, fore example, DFT. At present, such a comparison is still problematic, because the ab initio calculation of the many-polaron optical conductivity is much more complicated than the semianalytic treatment. It is not yet clear whether DFT is capable of providing the optical response for many-polaron systems. In the recent review [19], the Kubo formula for a many-polaron optical conductivity has been derived from the first principles. However, its practical application has not yet been fully realized. As noted in Ref. [19], \"in practice, the current-current correlation function is seldom evaluated exactly. Instead, it is common to work in the independent-particle approximation\", which is in fact equivalent to the single-polaron approach (maybe including many-polaron effects through corrections to single-polaron parameters). Moreover, the first-principle theory of the polaron response is at present still restricted to the DC mobility. Consequently, a comparison of our semianalytic approach with the first-principle calculations can be a subject of the future work.