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Contrary to the original expectation, Na$_2$IrO$_3$ is not a Kitaev's quantumspin liquid (QSL) but shows a zig-zag-type antiferromagnetic order inexperiments. Here we propose experimental clues and criteria to measure how amaterial in hand is close to the Kitaev's QSL state. For this purpose, wesystematically study thermal and spin excitations of a generalizedKitaev-Heisenberg model studied by Chaloupka $et$ $al$. in Phys. Rev. Lett.110, 097204 (2013) and an effective ab initio Hamiltonian for Na$_2$IrO$_3$proposed by Yamaji $et$ $al$. in Phys. Rev. Lett. 113, 107201 (2014), byemploying a numerical diagonalization method. We reveal that closeness to theKitaev's QSL is characterized by the following properties, besides trivialcriteria such as reduction of magnetic ordered moments and Neel temperatures:(1) Two peaks in the temperature dependence of specific heat at $T_{\ell}$ and$T_h$ caused by the fractionalization of spin to two types of Majoranafermions. (2) In between the double peak, prominent plateau or shoulder pinnedat $(R/2)\ln 2$ in the temperature dependence of entropy, where $R$ is the gasconstant. (3) Failure of the linear spin wave approximation at the low-lyingexcitations of dynamical structure factors. (4) Small ratio $T_{\ell}/T_h$close to or less than 0.03. According to the proposed criteria, Na$_2$IrO$_3$is categorized to a compound close to the Kitaev's QSL, and is proven to be apromising candidate for the realization of the QSL if the relevant materialparameters can further be tuned by making thin film of Na$_2$IrO$_3$ on varioussubstrates or applying axial pressure perpendicular to the honeycomb networksof iridium ions. Applications of these characterization to(Na$_{1-x}$Li$_x$)$_2$IrO$_3$ and other related materials are also discussed.

Clues and criteria for designing Kitaev spin liquid revealed by thermal and spin excitations of honeycomb iridates Na$_2$IrO$_3$