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Critical properties of the Kitaev-Heisenberg model

We study critical properties of the Kitaev-Heisenberg model on the honeycomb lattice at finite temperatures which might describe the physics of the quasi two-dimensional compounds, Na$_2$IrO$_3$ and Li$_2$IrO$_3$. The model undergoes two phase transitions as a function of temperature. At low temperature, thermal fluctuations induce magnetic long-range order by order-by-disorder mechanism. Magnetically ordered state with the spontaneously broken $Z_6$ symmetry persists up to a certain critical temperature. We find that there is an intermediate phase between the low-temperature ordered phase and the high-temperature disordered phase. The finite-sized scaling analysis suggests that the intermediate phase is a critical Kosterlitz-Thouless phase with continuously variable exponents. We argue that the intermediate phase has been actually observed above the low-temperature magnetically ordered phase in Na$_2$IrO$_3$, and likely in Li$_2$IrO$_3$.