Competition between $d$-wave and topological $p$-wave superconducting phases in the doped Kitaev-Heisenberg model

The competition between Kitaev and Heisenberg interactions away from half filling is studied for the hole-doped Kitaev-Heisenberg $t$-$J_K$-$J_H$ model on a honeycomb lattice. While the isotropic Heisenberg coupling supports a time-reversal violating $d$-wave singlet state, we find that the Kitaev interaction favors a time-reversal invariant $p$-wave superconducting phase, which obeys the rotational symmetries of the microscopic model, and is robust for $J_H<J_K/2$. Within the $p$-wave superconducting phase, a critical chemical potential $\left|\mu \right|={\mu }_c\approx t$ separates a topologically trivial phase for $\left|\mu \right|<{\mu }_c$ from a topologically nontrivial $Z_2$ time-reversal invariant spin-triplet phase for $\left|\mu \right|>{\mu }_c$.

Figure 1

(a) Relative orientations of the Cooper pair orbital (thick arrows) and spin (thin red/gray arrows) angular momentum in the ground-state wave function for topologically nontrivial time-reversal invariant $p$-wave pairing. Conventions for (b) real space and (c) reciprocal lattices. The four time-reversal invariant points in the Brillouin zone are $\Gamma$, $M_1$, $M_2$, and $M_3$.

Figure 2

The transition temperature $k_B{T}_c/t_0$ in the $p$-wave triplet channel as a function of the Kitaev interaction strength $J_K/t_0$ for different doping levels $\delta =0.1$, 0.25, 0.5, at $J_H=0$ (Ref. 30).

Figure 3

Phase diagram for the condensation in the spin-singlet (red/dark gray regions marked as $s$-SC and $d$-SC) and spin-triplet (green/light gray regions) channels as a function of Heisenberg interaction $J_H$ and doping $\delta$, for Kitaev interaction strength $J_K=t_0$. In the singlet channel both $d$-wave and $s$-wave order parameters can exist depending on the doping and interaction strength. The $d$-wave order parameter breaks the time-reversal symmetry. Within the $p$-wave triplet phase, there is a topological phase transition (indicated by a dashed line) between topologically trivial $\delta <0.25$ and topologically nontrivial $\delta >0.25$ time-reversal invariant superconducting states. The (blue/dark gray) region in the upper left corner of the phase diagram denotes the parameter regime where $k_B{T}_c<{10}^{-4}{t}_0$.

Figure 4

The minimum energy required to excite quasiparticles as a function of the dimensionless interband order parameter strength $\epsilon$ and chemical potential $\left|\mu \right|$ for $\eta =0.042t$. The interband order parameter is adiabatically switched on when $\epsilon$ changes from $\epsilon =0$ to $\epsilon =1$. The inset demonstrates that for each value of $\epsilon$ the energy gap closes in the vicinity of $\left|\mu \right|=t$, indicating a transition between the topologically trivial and nontrivial states.

Figure 5

Integrated density of states for $\eta =0.042t$ and chemical potentials $0.7t$, ${\mu }_c$, and $1.3t$. The critical value of the chemical potential is ${\mu }_c=0.9932t.$