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In this work we study the phase diagram of Kekul\'{e}-Kitaev model. The modelis defined on a honeycomb lattice with bond dependent anisotropic exchangeinteractions making it exactly solvable in terms of Majorana representation ofspins in close analogy to the Kitaev model. However, the energy spectrum ofMajorana fermions has a multi-band structure characterized by Chern numbers 0,$\pm$1, and $\pm2$. We obtained the phase diagram of the model in the plane ofexchange couplings and in the presence of a magnetic field and found chiraltopological and trivial spin-liquid ground states. In the absence of magneticfield most part of the phase diagram is a trivial gapped phase continuouslyconnected to an Abelian phase, while in the presence of the magnetic field atopological phase arises. Furthermore, motivated by recent thermal measurementson the spin-liquid candidate $\alpha$-RuCl$_{3}$, we calculated the thermalHall conductivity at different regimes of parameters and temperatures and foundthe latter is quantized over a wide range of temperatures.

Phase diagram and thermal Hall conductivity of spin-liquid Kekul\'{e}-Kitaev model