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

Contrary to the original expectation, Na$_2$IrO$_3$ is not a Kitaev's quantum spin liquid (QSL) but shows a zig-zag-type antiferromagnetic order in experiments. Here we propose experimental clues and criteria to measure how a material in hand is close to the Kitaev's QSL state. For this purpose, we systematically study thermal and spin excitations of a generalized Kitaev-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), by employing a numerical diagonalization method. We reveal that closeness to the Kitaev's QSL is characterized by the following properties, besides trivial criteria 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 Majorana fermions. (2) In between the double peak, prominent plateau or shoulder pinned at $(R/2)\ln 2$ in the temperature dependence of entropy, where $R$ is the gas constant. (3) Failure of the linear spin wave approximation at the low-lying excitations 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 a promising candidate for the realization of the QSL if the relevant material parameters can further be tuned by making thin film of Na$_2$IrO$_3$ on various substrates or applying axial pressure perpendicular to the honeycomb networks of iridium ions. Applications of these characterization to (Na$_{1-x}$Li$_x$)$_2$IrO$_3$ and other related materials are also discussed.