Critical Properties of the Kitaev-Heisenberg Model

We study the critical properties of the Kitaev-Heisenberg model on the honeycomb lattice at finite temperatures that might describe the physics of the quasi-two-dimensional compounds, ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and ${\mathrm{Li}}_2{\mathrm{IrO}}_3$. The model undergoes two phase transitions as a function of temperature. At low temperature, thermal fluctuations induce magnetic long-range order by the order-by-disorder mechanism. This magnetically ordered state with a 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. 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 observed above the low-temperature, magnetically ordered phase in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, and also, likely exists in ${\mathrm{Li}}_2{\mathrm{IrO}}_3$.

Figure 1

Four possible magnetic configurations: (a) the FM ordering; (b) the two-sublattice, AF Néel order; (c) the stripy order; (d) the zigzag order. Open and filled circles correspond to up and down directions of spins.

Figure 2

Histograms of the order parameter $m_{N(S)}$, obtained for the system with $2\times 84\times 84$ spins in the ordered phase, (a) and (e), in the intermediate phase, (b)–(c) and (f)–(g), and in the disordered phase, (d) and (h). Histograms (a)–(d) are computed for $\alpha =0.25$, and (e)–(h) are for $\alpha =0.75$. The histograms are presented on the complex plane (Re $|m_{N(S)}|$, Im $|m_{N(S)}|$).

Figure 3

A snapshot of the coarse-grained order parameter $⟨m_N⟩$ at $T=0.168$. The vortexlike topological excitations are evident.

Figure 4

The log-log plots of the order parameter $m_{N(S)}$ as a function of system size $L$ at various temperatures, which increase from the top to the bottom curves. The solid curves indicate the linear behavior that corresponds to a power law dependence, $m_{N(S)}\sim L^{-\eta /2}$, corresponding to the intermediate critical phase. The dashed curves show deviation away from the linear behavior outside the critical phase.

Figure 5

The Binder cumulant as a function of temperature for (a) $\alpha =0.25$ and (b) $\alpha =0.75$. From the crossing points of different Binder’s curves, we estimate $T_{c_1}=0.152$ and $T_{c_1}=0.124$ for $\alpha =0.25$ and $\alpha =0.75$, respectively. The curve corresponding to the largest system size ($L=204$) is the lowest at temperatures above the crossing point and the highest at temperatures below it.

Figure 6

Specific heat $C$ as a function of temperature for (a) $\alpha =0.25$ and (b) $\alpha =0.75$. The black line corresponds to the smallest system size ($L=36$), while the light grey line corresponds to the largest systems size ($L=204$).