Magnetic impurity affected by spin-orbit coupling: Behavior near a topological phase transition

We investigate the effect of spin-orbit coupling on the behavior of a magnetic impurity at the edge of a zigzag graphene ribbon by means of quantum Monte Carlo simulations. A peculiar interplay of Kane-Mele type spin-orbit and impurity-host coupling is found to greatly affect properties for the local moment. The local characters of the impurity are mainly dominated by the local density of states at the edge, such as double occupancy, magnetic moment, and spin susceptibilities. The special helical nature of the topological insulator on the boundary is found to affect nonlocal quantities, such as the two-particle and spin-spin correlation functions linking electrons on the impurity with those in the conduction band; in particular, due to the spin-orbit coupling, the symmetry of the spin rotation in the Kondo cloud around the impurity is partly broken.

Figure 1

(a) The LDOS as a function of energy $E/t$ with different values for $\lambda$. From top to bottom $\lambda =$ 0, $0.04t$, $0.08t$, and $0.2t$. (b) The spin-flip strength $J_{k-kl}^{\parallel }$ as a function of $\lambda$.

Figure 2

(a) Double occupancy $\langle n_{d\uparrow }{n}_{d\downarrow }\rangle$ versus chemical potential $\mu /t$. (b) $\langle {\left(S^z\right)}^2\rangle$ versus $\mu /t$. Here, $U=1.2t$, $V=0.65t$, ${\varepsilon }_d=-U/2$, and $T^{-1}=32t^{-1}$.

Figure 3

The spin susceptibility $\chi$ as a function of $T$ for different values of the SOC. $U=1.2t$, $V=0.65t$, ${\varepsilon }_d=-U/2$, and $\mu =0$. The solid line presents the results for the ZBWA model.

Figure 4

(a) The spin-flip processes in the normal states. (b) The spin-flip processes on the helical edges. $|\pm k,\uparrow \downarrow \rangle$ are the states of conduction electrons and up and down arrows mean local spins.

Figure 5

(a) Correlation function for backscattering and spin flip, $\langle c_{-k,\uparrow }{d}_{\uparrow }^{†}{d}_{\downarrow }{c}_{k,\downarrow }^{†}\rangle$ versus $\lambda$; (b) double occupancy as a function of $\lambda$. In both panels $V=0.65t$, ${\varepsilon }_d=-U/2$, $\mu =0$, $\beta =32t^{-1}$, and momentum $k=0.00125\pi$.

Figure 6

Spin-spin correlation functions $S_{\mathbf{i}}^z$ and $S_{\mathbf{i}}^x$ versus the index $\mathbf{i}$ at the edge, $U=1.2t=-2{\varepsilon }_d$, and $\beta =32t^{-1}$. In (a) $\mu =0$ and in (b) $\mu =-0.2t$ and $\lambda =0.15t$.

Figure 7

(a) The ratio $S_{\mathbf{i}}^x/S_{\mathbf{i}}^z$ with $\mu =-0.1t$. (b) The ratio $S_{\mathbf{i}}^x/S_{\mathbf{i}}^z$ with $\lambda =0.15t$ varied. In both panels $U=1.2t=-2{\varepsilon }_d$, and $\beta =32t^{-1}$.

Figure 8

Schematics of extracting the Fermi vectors $k_F$ from the band of $H_{\mathrm{K}-\mathrm{M}}$. (a) $\mu$ is fixed at $-0.1t$ and $\lambda =0.1t,0.15t$, and $0.2t$. (b) $\lambda$ is fixed at $0.15t$ and $\mu =0,-0.1t,-0.2t$, and $-0.5t$. Solid lines represent the bands for the Kane-Mele Hamiltonian.

Figure 9

Comparing $\varphi$ to the Fermi vectors $k_F$. (a) is for Figs. 7a and 8a and (b) is for Figs. 7b and 8b.

Figure 10

The ratio $S_{\mathbf{i}}^x/S_{\mathbf{i}}^z$ in the case of $\mu =-0.2t$ and $\lambda =0.15t$ with $U$ varied. $\beta =32t^{-1}$ and ${\varepsilon }_d$ is fixed at $-U/2$.