Anderson Localization from the Berry-Curvature Interchange in Quantum Anomalous Hall Systems

We theoretically investigate the localization mechanism of the quantum anomalous Hall effect (QAHE) in the presence of spin-flip disorders. We show that the QAHE stays quantized at weak disorders, then enters a Berry-curvature mediated metallic phase at moderate disorders, and finally goes into the Anderson insulating phase at strong disorders. From the phase diagram, we find that at the charge neutrality point although the QAHE is most robust against disorders, the corresponding metallic phase is much easier to be localized into the Anderson insulating phase due to the interchange of Berry curvatures carried, respectively, by the conduction and valence bands. In the end, we provide a phenomenological picture related to the topological charges to better understand the underlying physical origin of the QAHE Anderson localization.

Figure 1

Averaged conductance $⟨G⟩$ as a function of $W$ for different Fermi energies inside the bulk gap. The system width is set to be $N=80a$. (a) and (b) For nonmagnetic and spin-flip disorders, respectively. Over 3000 samples are collected for each point. Inset: Band structure of a nanoribbon displaying the gapless chiral edge states inside the bulk gap $\mathrm{\Delta }$.

Figure 2

($a_1$)–($h_1$): Evolution of averaged Berry curvature density $\mathrm{\Omega }$ as a function of energy $E$ for different disorder strengths $W$. The green and pink areas highlight the exchange process of Berry curvatures carried, respectively, by conduction and valence bands. ($a_2$)–($h_2$): Corresponding averaged Hall conductance ${\sigma }_{xy}$ as a function of energy $E$. The supercell is set to be $50a\times 50a$. Over 30 samples are collected for each point.

Figure 3

(a) Normalized localization length $\xi /L$ as a function of the disorder strength $W$ at $E_F=0.0$ calculated on quasi-one-dimensional bars, with a length of $2\times 10^6$ and different widths of $L=32$, 48, 64, and 96. $W_{c1}=3.18$ and $W_{c2}=3.23$ are two critical points. (b) Phase diagram in ($E$, $W$) plane.

Figure 4

Schematic of the evolution of topological charges carried by the spin textures (i.e., Skyrmions and merons) of the valence and conduction bands. (a) In the absence of disorders, valence or conduction bands carry a Skyrmion with a topological charge of $Q_v=-1/Q_c=+1$. (b) At weak disorders, the states near the equator of Skyrmions scatter strongly to divide Skyrmions into merons that carry half-integer topological charges and are labeled as “$A$,” “$B$,” “$C$,” and “$D$.” (c) By further increasing disorder strength, merons $A$ and $D$ respectively move upwards and downwards, while merons $B$ and $C$ move towards each other. (d) At even stronger disorders, merons $B$ and $C$ make an exchange to cancel out the topological charges in the new valence (conduction) bands by combing merons $A$ and $C$ ($B$ and $D$).