Coexistence of Quantum Hall and Quantum Anomalous Hall Phases in Disordered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$

In most cases, to observe quantized Hall plateaus, an external magnetic field is applied in intrinsic magnetic topological insulators ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. Nevertheless, whether the nonzero Chern number ($C\ne 0$) phase is a quantum anomalous Hall (QAH) state, or a quantum Hall (QH) state, or a mixing state of both is still a puzzle, especially for the recently observed $C=2$ phase [Deng et al., Science 367, 895 (2020)]. In this Letter, we propose a physical picture based on the Anderson localization to understand the observed Hall plateaus in disordered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. Rather good consistency between the experimental and numerical results confirms that the bulk states are localized in the absence of a magnetic field and a QAH edge state emerges with $C=1$. However, under a strong magnetic field, the lowest Landau band formed with the localized bulk states, survives disorder, together with the QAH edge state, leading to a $C=2$ phase. Eventually, we present a phase diagram of a disordered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ which indicates more coexistence states of QAH and QH to be verified by future experiments.

Figure 1

(a) A schematic plot of a six-terminal disordered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ device under a strong magnetic field. The wave packets represent the localized bulk states. In the absence of a magnetic field, bulk states are localized by disorder and the QAH edge state emerges. For a strong magnetic field, these localized bulk states can still form Landau bands coexisting with the QAH state. (b) Calculated Hall resistance $R_{yx}$ as a function of magnetic field $\mathrm{\Phi }/{\varphi }_0$ under different Fermi energies. (c) Calculated longitudinal resistance $R_{xx}$ and $R_{yx}$ as a function of Fermi energy. The insets show the experimental measured $R_{xx}$ and $R_{yx}$ adopted from Ref. [3], where ${\mu }_{0}H$ and $V_g$ correspond to the magnetic field and Fermi energy, respectively. The numerical simulation is performed with a sample size of $400\times 400$ sites and averaged over 320 disorder configurations.

Figure 2

(a),(b) The ratio of the geometric mean DOS ${\rho }_{\mathrm{ave}}$ to the arithmetic mean DOS ${\rho }_{\mathrm{typ}}$ with $\mathrm{\Phi }/{\varphi }_0=0$ and $\mathrm{\Phi }/{\varphi }_0=0.06$, respectively. The inset in (b) shows the IPR $P^{-1}$ with $E_F=0.27$. (c) and (d) show ${\rho }_{\mathrm{ave}}$ with $\mathrm{\Phi }/{\varphi }_0=0$ and $\mathrm{\Phi }/{\varphi }_0=0.06$, respectively. (e),(f) $P^{-1}$ with $E_F=0.08$ and $E_F=0.33$ in the absence of magnetic field, separately. $N=M^2$ is the total sites of the sample. The blue and red points are from numerical simulation, and the black line represents a linear fit to the data. All the data are calculated with $N_x=N_y=M$.

Figure 3

(a),(b) Renormalized localization length ${\mathrm{\Lambda }}_M=\lambda (M)/M$ against the magnetic field $\mathrm{\Phi }/{\varphi }_0$ with different Fermi energies. (c),(d) ${\mathrm{\Lambda }}_M$ as a function of Fermi energy under different magnetic field strengths. The calculation is performed with the periodic boundary condition in the $x$ direction.

Figure 4

Phase diagram of a disordered ferromagnetic insulator ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ in the $E_F-\mathrm{\Phi }/{\varphi }_0$ plane. The blue balls mark the extended states of Landau bands. The phase diagram is identified from the renormalized localization length ${\mathrm{\Lambda }}_M$.