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We use an exact numerical diagonalization method to calculate the dynamicalspin structure factors (DSFs) of three ab-initio models and oneab-initio-guided model for a honeycomb-lattice magnet $\alpha$-RuCl$_3$. Wealso use thermal pure quantum states to calculate the temperature dependence ofthe heat capacity, the nearest-neighbor (NN) spin-spin correlation function,and the static spin structure factor. From the results obtained from these foureffective models, we find that, even when the magnetic order is stabilized atlow temperature, the intensity at the $\Gamma$ point in the DSFs increases withincreasing NN spin correlation. In addition, we find that the four models failto explain heat-capacity measurements whereas two of the four models succeed inexplaining inelastic-neutron-scattering (INS) experiments. In the four models,when temperature decreases, the heat capacity shows a prominent peak at a hightemperature where the NN spin-spin correlation function increases. However, thepeak temperature in heat capacity is too low in comparison with that observedexperimentally. To address these discrepancies, we propose an effective modelthat includes strong ferromagnetic Kitaev coupling, and we show that this modelquantitatively reproduces both INS experiments and heat-capacity measurements.To further examine the adequacy of the proposed model, we calculate the fielddependence of the polarized terahertz spectra, which reproduces theexperimental results: the spin-gapped excitation survives up to an onset fieldwhere the magnetic order disappears and the response in the high-field regionis almost linear. Based on these numerical results, we argue that thelow-energy magnetic excitation in $\alpha$-RuCl$_3$ is mainly characterized byinteractions such as off-diagonal interactions and weak Heisenberg interactionsbetween NN pairs, rather than by the strong Kitaev interactions.

Effective model with strong Kitaev interactions for $\alpha$-${\rm RuCl_3}$