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