Order by disorder in Kitaev-Heisenberg models on the honeycomb lattice*

Recent diffuse magnetic x-ray scattering data in Na2IrO3 [1] clearly determined the spin orientation in this zigzag state and showed that, unexpectedly, it is along  the 44.3 degrees direction  with respect to a axis, which is approximately half way in between the cubic x and y axes.  This experiment provides an important check of the validity of any  model proposed to described the  magnetic properties  of Na2IrO3 as the model should correctly predict not only the type of the magnetic order but also  its orientation in space. We propose that order by disorder mechanism in quantum J1-K1-J2-K2-J3 model [2] gives the experimentally observed direction along cubic face diagonals. Our findings are based on both the calculation of the contribution  of thermal fluctuations of quantum spins into free energy   obtained by Hubbard-Stratonovich transformation and  the zero-point correction to the ground state energy due to quantum spin fluctuations obtained by the spin-wave expansion at zero temperature. [1] S. H. Chun et al, Nature Physics10,1038 (2015).? [2] Y. Sizyuk, C. Price, P. Woelfle, and N. B. Perkins,? Phys. Rev. B  90, 155126 (2014).