  We report the temperature-dependent optical conductivity and ARPES studies of
the iron-based superconductor (SC) Sr$_{0.67}$Na$_{0.33}$Fe$_2$As$_2$ in the
high-temperature tetragonal paramagnetic phase; below the structural and
magnetic transitions at $T_{\rm N}\simeq$125 K in the orthorhombic
spin-density-wave (SDW)-like phase, and $T_r\simeq$42 K in the reentrant
tetragonal double-Q magnetic phase where both charge and SDW order exist; and
below the SC transition at $T_c\simeq$10 K. The free-carrier component in the
optical conductivity is described by two Drude contributions; one strong and
broad, the other weak and narrow. The broad Drude component decreases
dramatically below $T_{\rm N}$ and $T_r$, with much of its strength being
transferred to a bound excitation in the mid-infrared, while the narrow Drude
component shows no anomalies at either of the transitions, actually increasing
in strength at low temperature while narrowing dramatically. The behavior of an
infrared-active mode suggests zone-folding below $T_r$. Below $T_c$ the
dramatic decrease in the low-frequency optical conductivity signals the
formation of a SC energy gap. ARPES reveals hole-like bands at the center of
the Brillouin zone (BZ), with both electron- and hole-like bands at the
corners. Below $T_{\rm N}$, the hole pockets at the center of the BZ decrease
in size, consistent with the behavior of the broad Drude component; while below
$T_r$ the electron-like bands shift and split, giving rise to a low-energy
excitation in the optical conductivity at ~20 meV. The magnetic states, with
resulting SDW and charge-SDW order, respectively, lead to a significant
reconstruction of the Fermi surface that has profound implications for the
transport originating from the electron and hole pockets, but appears to have
relatively little impact on the SC in this material.
