Microscopic theory of refractive index applied to metamaterials: Effective current response tensor corresponding to standard relation $n^2 = \varepsilon_{\mathrm{eff}} \mu_{\mathrm{eff}}$

In this article, we first derive the wavevector- and frequency-dependent, microscopic current response tensor which corresponds to the "macroscopic" ansatz $\vec D = \varepsilon_0 \varepsilon_{\mathrm{eff}} \vec E$ and $\vec B = \mu_0 \mu_{\mathrm{eff}} \vec H$ with wavevector- and frequency-independent, "effective" material constants $\varepsilon_{\mathrm{eff}}$ and $\mu_{\mathrm{eff}}$. We then deduce the electromagnetic and optical properties of this effective material model by employing exact, microscopic response relations. In particular, we argue that for recovering the standard relation $n^2 = \varepsilon_{\mathrm{eff}} \mu_{\mathrm{eff}}$ between the refractive index and the effective material constants, it is imperative to start from the microscopic wave equation in terms of the transverse dielectric function, $\varepsilon_{\mathrm{T}}(\vec k, \omega) = 0$. On the phenomenological side, our result is especially relevant for metamaterials research, which draws directly on the standard relation for the refractive index in terms of effective material constants. Since for a wide class of materials the current response tensor can be calculated from first principles and compared to the model expression derived here, this work also paves the way for a systematic search for new metamaterials.