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Finite temperature phase diagram of the classical Kitaev-Heisenberg model

We investigate the finite-temperature phase diagram of the classical Kitaev-Heisenberg model on the hexagonal lattice. Due to the anisotropy introduced by the Kitaev interaction, the model is magnetically ordered at low temperatures for all values of parameters at which the model has a discrete symmetry. The ordered phase is stabilized entropically by an order by disorder mechanism where thermal fluctuations of classical spins select collinear magnetic orders in which magnetic moments point along one of the cubic directions. We find that there is an intermediate phase between the low-temperature ordered phase and the high-temperature disordered phase. We show that the intermediate phase is a critical Kosterlitz-Thouless phase exhibiting correlations of the order parameter that decay algebraically in space. Using finite size scaling analysis, we determine the boundaries of the critical phase with reasonable accuracy. We show that the Kitaev interaction plays a crucial role in understanding the finite temperature properties of A$_2$IrO$_3$ systems.