# Optical Response of an Interacting Polaron Gas in Strongly Polar Crystals
## 3 Semianalytic Approximations  
The optical conductivity of an interacting polaron gas is calculated here within the extended random phase approximation (RPA), as described below. The memory function \(\chi\left(\Omega\right)\) can be expressed through the polarization function of the electron gas for sufficiently small \(\mathbf{q}\). Thus, the RPA can be applied under the assumption that the long-wavelength phonons bring the dominant contribution to the polaron optical response. In this approximation, the band energy \(\varepsilon_{\lambda}\left(\mathbf{k}\right)\) which enters the operator \(\hat{\beta}_{\mathbf{q}}\) is expanded in powers of the momentum as \(\varepsilon_{\lambda}\left(\mathbf{k}\right)=\hbar^{2}k^{2}/2m_{b,\lambda}+O \left(k\right)^{4}\). In SrTiO\({}_{3}\), the band mass obtained within the tight-binding analytic fit as described in Appendix A appears isotropic with the same value \(m_{b}\approx 0.72m_{e}\) (where \(m_{e}\) is the electron mass in vacuum) for all three subbands of the conduction band. Thus, the memory function for the optical conductivity is approximated by the expression:  
\[\chi\left(\Omega\right)=\frac{2e^{2}}{3\hbar m_{b}^{2}\mathcal{Z}}\sum_{ \mathbf{q},j}\left|V_{\mathbf{q},j}\right|^{2}q^{2}\int_{0}^{\infty}dt\,e^{- \delta t}\left(e^{i\Omega t}-1\right)\text{Im}\left[T^{*}\left(\omega_{ \mathbf{q},j},t\right)\left\langle\rho_{\mathbf{q}}\left(t\right)\rho_{ \mathbf{q}}^{\dagger}\right\rangle_{0}\right], \tag{17}\]  
where \(\rho_{\mathbf{q}}=\sum_{\mathbf{k},\sigma,\lambda}a_{\mathbf{k}+\mathbf{q}, \sigma,\lambda}^{\dagger}a_{\mathbf{k},\sigma,\lambda}\) is the Fourier component of the electron density. Please note that below, when calculating the density-density correlation function, we do not perform this expansion. The treatment is restricted here to the optical response provided by the Frohlich electron-phonon interaction with LO phonons, because this mechanism brings the main contribution to the mid-infrared optical conductivity in polar crystals. The LO- and TO-phonon frequencies are denoted below as \(\omega_{\mathbf{q},j}\equiv\omega_{L,j}\left(\mathbf{q}\right)\) and \(\omega_{T,j}\left(\mathbf{q}\right)\), respectively. In the present work, they are not calculated, being input parameters of the theory. We use the Frohlich interaction amplitudes with the partial coupling constants \(\alpha_{j}\) for the \(j\)-th LO-phonon branch in a multimode polar crystals as derived in Ref. [13]:  
\[V_{\mathbf{q},j}=\frac{\hbar\omega_{L,j}}{q}\left(\frac{4\pi\alpha_{j}}{V} \right)^{1/2}\left(\frac{\hbar}{2m_{b}\omega_{L,j}}\right)^{1/4}, \tag{18}\]  
where \(\omega_{L,j}\equiv\left.\omega_{L,j}\left(\mathbf{q}\right)  
with  
\[\mathcal{G}\left(\mathbf{q},\Omega\right) \equiv-i\int_{0}^{\infty}e^{i\Omega t}\left\langle\rho_{\mathbf{q}} \left(t\right)\rho_{-\mathbf{q}}\left(0\right)\right\rangle_{0}\;dt, \tag{20}\] \[G^{R}\left(\mathbf{q},\Omega\right) \equiv-i\int_{0}^{\infty}e^{i\Omega t}\left\langle\left[\rho_{ \mathbf{q}}\left(t\right),\rho_{-\mathbf{q}}\left(0\right)\right]\right\rangle _{0}\;dt. \tag{21}\]  
In Ref. [13], the Green's functions were calculated within the RPA for an electron gas. Here, we apply the RPA extended for an interacting electron-phonon system, which leads to a formula structurally similar to that obtained within RPA, but with a different (momentum and frequency dependent) electron-electron interaction matrix element:  
\[\frac{4\pi e^{2}}{\varepsilon_{\infty}q^{2}}\rightarrow\frac{4\pi e^{2}}{ \varepsilon_{L}\left(\mathbf{q},\omega\right)q^{2}}, \tag{22}\]  
with the dielectric function of the lattice \(\varepsilon_{L}\left(\mathbf{q},\omega\right)\), which describes the dynamic lattice polarization.