  Combining thermodynamic measurements with theoretical density functional and
thermodynamic calculations we demonstrate that the honeycomb lattice iridates
A2IrO3 (A = 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 Heisenberg interactions beyond
nearest-neighbor exchange. Experimentally, we observe an increase of the
Curie-Weiss temperature from \theta = -125 K for Na2IrO3 to \theta = -33 K for
Li2IrO3, while the antiferromagnetic ordering temperature remains roughly the
same T_N = 15 K for both materials. Using finite-temperature functional
renormalization group calculations we show that this evolution of \theta, T_N,
the frustration parameter f = \theta/T_N, and the zig-zag magnetic ordering
structure suggested for both materials by density functional theory can be
captured within this extended HK model. Combining our experimental and
theoretical results, we estimate that Na2IrO3 is deep in the magnetically
ordered regime of the HK model (\alpha \approx 0.25), while Li2IrO3 appears to
be close to a spin-liquid regime (0.6 < \alpha < 0.7).
