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We study the effects of doping the Kitaev model on the honeycomb latticewhere the spins interact via the bond-directional interaction $J_K$, which isknown to have a quantum spin liquid as its exact ground state. The effect ofhole doping is studied within the $t$-$J_K$ model on a three-leg cylinder usingdensity-matrix renormalization group. Upon light doping, we find that theground state of the system has quasi-long-range charge-density-wavecorrelations but short-range single-particle correlations. The dominant pairingchannel is the even-parity superconducting pair-pair correlations with$d$-wave-like symmetry, which oscillate in sign as a function of separationwith a period equal to that of the spin-density wave and two times thecharge-density wave. Although these correlations fall rapidly (possiblyexponentially) at long distances, this is never-the-less the first examplewhere a pair-density wave is the strongest SC order on a bipartite lattice. Ourresults may be relevant to ${\rm Na_2IrO_3}$ and $\alpha$-${\rm RuCl_3}$ upondoping.

Evidence of pair-density wave in doping Kitaev spin liquid on the honeycomb lattice