Abstract

Combining thermodynamic measurements with theoretical calculations we demonstrate that the iridates ${A}_{2}{\mathrm{IrO}}_{3}$ ($A=\mathrm{Na}$, Li) are magnetically ordered Mott insulators where the magnetism of the effective spin-orbital $S=1/2$ moments can be captured by a Heisenberg-Kitaev (HK) model with interactions beyond nearest-neighbor exchange. Experimentally, we observe an increase of the Curie-Weiss temperature from $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}125\text{ }\text{ }\mathrm{K}$ for ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ to $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}33\text{ }\text{ }\mathrm{K}$ for ${\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, while the ordering temperature remains roughly the same ${T}_{N}\ensuremath{\approx}15\text{ }\text{ }\mathrm{K}$. Using functional renormalization group calculations we show that this evolution of $\ensuremath{\theta}$ and ${T}_{N}$ as well as the low temperature zigzag magnetic order can be captured within this extended HK model. We estimate that ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ is deep in a magnetically ordered regime, while ${\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ appears to be close to a spin-liquid regime.