Effects of Random Domains on the Zero Hall Plateau in the Quantum Anomalous Hall Effect

Recently, a zero Hall conductance plateau with random domains was experimentally observed in the quantum anomalous Hall (QAH) effect. We study the effects of random domains on the zero Hall plateau in QAH insulators. We find that the structure inversion symmetry determines the scaling property of the zero Hall plateau transition in the QAH systems. In the presence of structure inversion symmetry, the zero Hall plateau state shows a quantum-Hall-type critical point, originating from the two decoupled subsystems with opposite Chern numbers. However, the absence of structure inversion symmetry leads to a mixture between these two subsystems, gives rise to a line of critical points, and dramatically changes the scaling behavior. Hereinto, we predict a Berezinskii-Kosterlitz-Thouless-type transition during the Hall conductance plateau switching in the QAH insulators. Our results are instructive for both theoretic understanding of the zero Hall plateau transition and future transport experiments in the QAH insulators.

Figure 1

(a) Schematic plot of zero Hall plateau state on a $48\times 48$ magnetic TI lattice with equal population of up ($+$) and down ($-$) magnetic domains in unit of $8\times 8$ supercell. The black and red arrows represent the edge channels propagating on the two domains species with up ($+$) and down ($-$) magnetization. Due to the structural inversion symmetry, the system can be divided into two subsystems (b) and (c) describing two domain species, respectively. There is an effective chiral edge channel (dash line) in each of the subsystems. (d) The two subsystems are topologically equivalent to two Chern insulators with opposite Chern numbers $C_{\pm }=\pm 1$. Therefore, the whole system is topologically nontrivial with a quantized spin-Chern number $C_s\equiv (C_{+}-C_{-})/2$. Here $E_F$ is the Fermi energy and $C_{\pm }$ are averaged over 20 random-domain configurations with the sample size $96\times 96$.

Figure 2

(a) plots renormalized localization length $\mathrm{\Lambda }$ against diagonal disorder strength $W$ (in $V_d$) for $h_{+}$ with up ($+$) domains as shown in Fig. 1. (b) shows single parameter scaling of $\mathrm{\Lambda }$. All the data on the right side of the critical point collapse to a single curve by a scaling function $\mathrm{\Lambda }=f(L/\xi )$ with the correlation length $\xi$. (c) plots $\ln \xi$ against $-\ln |W-W_c|$, which can be fitted with the critical exponent (slope) $\nu =2.39\pm 0.19$ and critical disorder strength $W_c=2.70\pm 0.06$. (d) Chern number $C_{+}$ decreases from one to zeros with increasing disorder strength $W$. The behavior of $C_{-}$ is similar to $C_{+}$. Fermi energy $E_{\mathrm{F}}=0.1$ and $C\pm$ are averaged over 40 disordered configurations with sample size $96\times 96$.

Figure 3

(a) The phase diagram on the plane of Fermi energy $E_{\mathrm{F}}$ and SIA disorder strength $W$ in $V_A$. The dashed line corresponds to the parameter values studied in (b)–(d). (b) shows the renormalized localization length $\mathrm{\Lambda }$ versus $W$. (c) shows single parameter scaling of $\mathrm{\Lambda }$. Logarithmic correlation length $\ln \xi$ is fitted with linearly function of $1/\sqrt{|W-W_c|}$ with $W_c=13.4\pm 0.3$, indicating the BKT-type phase transition. The data on the right side of the critical point collapse to a single curve in the inset. (d) plots Chern numbers $C_{\pm }$ of two-subsystem spaces as a function of SIA disorder strength $W$, correspondingly. SIA potential difference $U_A=0$ and $C_{\pm }$ are averaged over 40 disordered configurations with sample size $96\times 96$.

Figure 4

(a) Phase diagram of magnetic TI with random domains on the plane of diagonal disorder strength $W$ for $V_d$ and spatial-averaged normalized magnetization $⟨M_z(\mathbf{r})⟩/M_s$. The spin Chern insulator (spin CI), Chern insulator (CI), and normal insulator (NI) are separated by a critical point or a critical region. The symbols guided by the solid and dash lines are obtained from localization length scaling for $U_A=0.05$ and $U_A=0$ with $V_A=0$, respectively. The dotted line corresponds to the parameter values in (b), which shows disorder-averaged Hall conductance ${\sigma }_{xy}$ and renormalized localization length $\mathrm{\Lambda }$ as a function of $⟨M_z(\mathbf{r})⟩/M_s$. The inset shows $\ln \xi$ versus $1/\sqrt{|W-W_c|}$ with $W_c=0.189\pm 0.004$. Other parameters are similar to those in Fig. 3.