Global phase diagram of a doped Kitaev-Heisenberg model

The global phase diagram of a doped Kitaev-Heisenberg model is studied using an $SU\left(2\right)$ slave-boson mean-field method. Near the Kitaev limit, $p$-wave superconducting states which break the time-reversal symmetry are stabilized as reported by You et al. [Phys. Rev. B 86, 085145 (2012)] irrespective of the sign of the Kitaev interaction. By further doping, a $d$-wave superconducting state appears when the Kitaev interaction is antiferromagnetic, while another $p$-wave superconducting state appears when the Kitaev interaction is ferromagnetic. This $p$-wave superconducting state does not break the time-reversal symmetry as reported by Hyart et al. [Phys. Rev. B 85, 140510 (2012)], and such a superconducting state also appears when the antiferromagnetic Kitaev interaction and the ferromagnetic Heisenberg interaction compete. This work, thus, demonstrates the clear difference between the antiferromagnetic Kitaev model and the ferromagnetic Kitaev model when carriers are doped while these models are equivalent in the undoped limit, and how novel superconducting states emerge when the Kitaev interaction and the Heisenberg interaction compete.

Figure 1

Schematic view of the Kitaev-Heisenberg model. $\gamma =x,y,z$ in the left figure show the spin components for the Kitaev interaction. ${\vec{r}}_{x,y,z}$ are unit vectors connecting the nearest-neighbor sites. On the right, the first Brillouin zone is shown.

Figure 2

Lanczos exact diagonalization results for squared total spins (normalized to their values in the fully polarized FM state) and the NN spin correlations obtained on 24-site clusters as a function of $J_K$ with $J_H=\pm (1-|J_K\left|\right)$. (a) AFM Kitaev case $J_K>0$ and (b) FM Kitaev $J_K<0$. Solid (dashed) lines correspond to original (rotated) spin basis. Vertical dash-dotted lines are first-order phase boundaries. For each case, a magnetic pattern is schematically shown.

Figure 3

Mean-field phase diagrams for the doped Kitaev-Heisenberg model as a function of $\delta$ and $J_K$ with $J_H=\pm (2-|J_K\left|\right)$. (a) AFM Kitaev ($J_K>0$) and (b) FM Kitaev ($J_K<0$). The phase boundary between the trivial $p$ SC${}_2$ and the topological $p$ SC${}_2$ in (b) (light dash-dotted line) is obtained with reduced SC order parameters as discussed in the main text. We also plot phase boundaries at $\delta =0$ obtained from the exact diagonalization as open stars.

Figure 4

Dispersion relations of the Majorana modes in $p$ SC${}_1$ phases. Upper figures: undoped Kitaev-Heisenberg model; and lower figures: doped Kitaev-Heisenberg model. Parameter values are indicated.

Figure 5

Comparison between the $p$ SC${}_1$ and the $p$ SC${}_2$ for the FM Kitaev with $J_K=-2$ and $J_H=0$. The total energy $E$ (a) and order parameters (b) as a function of $\delta$. $t_2$ is defined in Eq. (20). The light vertical line in (a) indicates the boundary between the $p$ SC${}_1$ phase and the $p$ SC${}_2$ phase.

Figure 6

Comparison between the $p$ SC${}_1$ and the $d+id$ SC for the AFM Kitaev with $J_K=2$ and $J_H=0$. The total energy $E$ (a) and order parameters (b) as a function of $\delta$. The light vertical line in (a) indicates the boundary between the $p$ SC${}_1$ phase and the $d+id$ SC phase.

Figure 7

$SU\left(2\right)$ SBMF results for the $p$ SC${}_2$ phase in the doped AFM Kitaev–FM Heisenberg model with $J_K=1.3$ and $J_H=-0.7$. (a) Order parameters and (b) SC gap amplitude as a function of doping concentration $\delta$.

Figure 8

$SU\left(2\right)$ SBMF results for the $p$ SC${}_2$ phase in the doped FM Kitaev model with $J_K=-2$ and $J_H=0$. (a) Order parameters and (b) SC gap amplitude as a function of doping concentration $\delta$. In (b), gap amplitudes obtained by using artificially reduced SC order parameters as $\langle t_{\rho }^{\gamma }\rangle \Rightarrow r\langle t_{\rho }^{\gamma }\rangle$ with $r<1$ are also shown with various $r$ indicated. For $r\lesssim 0.4$, there appear two gap minima, indicating the sequence of transitions from the even-parity trivial phase (small $\delta$) to the odd-parity trivial phase (intermediate $\delta$) and to the odd-parity topological phase (large $\delta$).

Figure 9

Comparison between the $d+id$ SC and the $s$ SC for the AFM Kitaev–AFM Heisenberg model with $J_K=0.8$ and $J_H=1.2$. The total energy $E$ (a) and order parameters (b) as a function of $\delta$. The light vertical line in (a) indicates the boundary between the $d+id$ SC phase and the $s$ SC phase.