Specific heat of Ca$_{0.32}$Na$_{0.68}$Fe$_2$As$_2$ single crystals: unconventional s$_\pm$ multi-band superconductivity with intermediate repulsive interband coupling and sizable attractive intraband couplings

We report a low-temperature specific heat study of high-quality single crystals of the heavily hole doped superconductor Ca$_{0.32}$Na$_{0.68}$Fe$_2$As$_2$. This compound exhibits bulk superconductivity with a transition temperature $T_c \approx 34$\,K, which is evident from the magnetization, transport, and specific heat measurements. The zero field data manifests a significant electronic specific heat in the normal state with a Sommerfeld coefficient $\gamma \approx 53$ mJ/mol K$^{2}$. Using a multi-band Eliashberg analysis, we demonstrate that the dependence of the zero field specific heat in the superconducting state is well described by a three-band model with an unconventional s$_\pm$ pairing symmetry and gap magnitudes $\Delta_i$ of approximately 2.35, 7.48, and -7.50 meV. Our analysis indicates a non-negligible attractive intraband coupling,which contributes significantly to the relatively high value of $T_c$. The Fermi surface averaged repulsive and attractive coupling strengths are of comparable size and outside the strong coupling limit frequently adopted for describing high-$T_c$ iron pnictide superconductors. We further infer a total mass renormalization of the order of five, including the effects of correlations and electron-boson interactions.