High-temperature thermodynamics of the honeycomb-lattice Kitaev-Heisenberg model: A high-temperature series expansion study

We develop high-temperature series expansions for the thermodynamic properties of the honeycomb-lattice Kitaev-Heisenberg model. Numerical results for uniform susceptibility, heat capacity, and entropy as a function of temperature for different values of the Kitaev coupling $K$ and Heisenberg exchange coupling $J$ (with $\left|J\right|\le \left|K\right|$) are presented. These expansions show good convergence down to a temperature of a fraction of $K$ and in some cases down to $T=K/10$. In the Kitaev exchange dominated regime, the inverse susceptibility has a nearly linear temperature dependence over a wide temperature range. However, we show that already at temperatures ten times the Curie-Weiss temperature, the effective Curie-Weiss constant estimated from the data can be off by a factor of 2. We find that the magnitude of the heat-capacity maximum at the short-range-order peak, is substantially smaller for small $J/K$ than for $J$ of order or larger than $K$. We suggest that this itself represents a simple marker for the relative importance of the Kitaev terms in these systems. Somewhat surprisingly, both heat-capacity and susceptibility data on ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ are consistent with a dominant antiferromagnetic Kitaev exchange constant of about $300–400\text{K}$.

Figure 1

A plot of the uniform susceptibility of the model as a function of temperature $T$ for different values of exchange constants $K$ and $J$. The upper plot corresponds to an antiferromagnetic Kitaev coupling and the lower plot to a ferromagnetic Kitaev coupling.

Figure 2

A plot of the inverse susceptibility of the model as a function of temperature $T$ for different values of exchange constants $K$ and $J$. The linear temperature dependence of the inverse susceptibility seemingly extends well below $T=K$.

Figure 3

The effective Curie-Weiss $T_{cw}^{\mathrm{eff}}$ constant obtained by a straight-line extrapolation of the inverse susceptibility locally from some temperature $T$ as a function of temperature $T$. Note that in the Kitaev regime the effective Curie-Weiss constant can differ from the true Curie-Weiss $T_{cw}$ constant by a factor of 2 already at a temperature ten times the Curie-Weiss temperature.

Figure 4

A plot of the entropy per mole of the model as a function of temperature $T$ for different values of the Heisenberg coupling $J$ and the Kitaev coupling $K$. The upper plot corresponds to an antiferromagnetic Kitaev coupling and the lower plot to a ferromagnetic Kitaev coupling. Some of the Padé approximants have nearby pole-zero pair around some temperature, that shows as a sharp glitch in the plots. These approximants do not converge near that temperature, but should be fine away from that temperature.

Figure 5

A plot of the heat capacity per mole of the model in units of the gas constant $R$ as a function of temperature $T$ for different values of the Heisenberg coupling $J$ and the Kitaev coupling $K$. The upper plot corresponds to an antiferromagnetic Kitaev coupling and the lower plot to a ferromagnetic Kitaev coupling. Note that a forklike split of approximants shows that the convergence is breaking down at lower temperatures. For small Heisenberg couplings, the convergence is better for ferromagnetic Kitaev coupling than for antiferromagnetic Kitaev coupling.