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

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