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We report a low-temperature specific heat study of high-quality singlecrystals of the heavily hole doped superconductorCa$_{0.32}$Na$_{0.68}$Fe$_2$As$_2$. This compound exhibits bulksuperconductivity with a transition temperature $T_c \approx 34$\,K, which isevident from the magnetization, transport, and specific heat measurements. Thezero field data manifests a significant electronic specific heat in the normalstate with a Sommerfeld coefficient $\gamma \approx 53$ mJ/mol K$^{2}$. Using amulti-band Eliashberg analysis, we demonstrate that the dependence of the zerofield specific heat in the superconducting state is well described by athree-band model with an unconventional s$_\pm$ pairing symmetry and gapmagnitudes $\Delta_i$ of approximately 2.35, 7.48, and -7.50 meV. Our analysisindicates a non-negligible attractive intraband coupling,which contributessignificantly to the relatively high value of $T_c$. The Fermi surface averagedrepulsive and attractive coupling strengths are of comparable size and outsidethe strong coupling limit frequently adopted for describing high-$T_c$ ironpnictide superconductors. We further infer a total mass renormalization of theorder of five, including the effects of correlations and electron-bosoninteractions.

physical review. b, condensed matter and materials physics

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