Berry-Curvature Exchange Induced Anderson Localization in Large-Chern-Number Quantum Anomalous Hall Effect

In the bilayer graphene based QAHE system, we select several representative Fermi energies inside the bulk band gap, i.e., $E_{\rm F}/t=0.001,~{}0.04,~{}0.08,~{}0.12,~{}0.16,~{}0.20$ to investigate the disorder effect on the electronic transport properties. Figure 2 depicts the averaged two-terminal conductance $\langle G\rangle$ as a function of the disorder strength $W$ for respective nonmagnetic and spin-flip disorders. For the nonmagnetic case as displayed in Fig. 2(a), the conductance keeps quantized to be $\langle G\rangle=4e^{2}/h$ in the presence of weak disorders. When the disorder strength reaches certain critical strength, the conductance suddenly becomes to decrease first and then gradually decrease to be vanishing. In particular, at a fixed disorder strength, the closer of the Fermi energy near the charge neutrality point $E_{\rm F}/t=0.00$, the more robust of the averaged conductance against disorder. However, for the spin-flip disorder case, the disorder effect on the electronic transport properties exhibit completely different from those observed in the nonmagnetic case. Figure 2(b) shows the averaged conductance as a function of disorder strength in the presence of spin-flip disorder. One can find that the variation of the averaged conductance is divided into two parts with the increase of disorder strength, i.e. (i) in the presence of weak disorder $(W/t\textless 2.3)$, the averaged conductance keeps quantized ($\langle G\rangle=4~{}e^{2}/h$) before it reaches the critical disorder strength, (ii) after disorder exceeds the critical strength, the averaged conductance gradually vanishes. In contrast to the nonmagnetic case, the farther of the Fermi energy away from the charge neutrality point $E_{\rm F}/t=0.00$, the more difficult of the conductance to be destroyed. For instance, the averaged conductance for $E_{\rm F}/t=0.001$ keeps quantized until $(W/t\approx 2.2)$ and once $W/t>2.3$, it immediately drops to zero.

In the magnetic topological insulator thin films, Fig. 3 displays the averaged conductance as a function of disorder strength for different Fermi energies inside the bulk gap. In the presence of nonmagnetic disorder [see Fig. 3(a)], the variation of averaged conductance with the increase of disorder strength is similar with the nonmagnetic case of bilayer graphene system [see Fig. 2(a)]. While in the presence of spin-flip disorders, as shown in Fig. 3(b), the system exhibits a completely different localization phenomenon in contrast to the above numerical results. The most striking feature is the appearance of additional quantized conductance plateau as the increases of disorder strength $W$, i.e., in the presence of weak disorder, the averaged conductance keeps quantized with a value of $\langle G\rangle=2e^{2}/h$; when $W$ reaches the first critical value (e.g., $W_{\rm C1}/B\approx 1.90$ at $E_{\rm F}/B=0.001$), the initial quantized plateau is destroyed and the system quickly enters a lower quantized plateau with a value of $\langle G\rangle=e^{2}/h$; as $W$ continuing to increase, the averaged conductance keeps quantized at $\langle G\rangle=e^{2}/h$; when $W$ reaches the second critical point (e.g., $W_{\rm C2}/B\approx 3.90$ at $E_{\rm F}/B=0.001$), the quantized plateau is destroyed and the conductance gradually vanishes. Note that, the closer of the Fermi energy to $E_{\rm F}/B=0.00$, the more robust of the lower quantized plateau against disorder, therefore the lower quantized plateau almost disappears at the conduction band edge ($E_{\rm F}/B=0.20$).

For magnetic topological insulator thin films, the Chern number can be tuned by varying the thickness of the films. Inset of Fig. 3(b) shows the averaged conductance as a function of disorder strength for the system of ${\mathcal{C}}=4$ and ${\mathcal{C}}=3$ at $E_{\rm F}/B=0.001$. One can see that the Hall conductance is destroyed step by step and forms plateaus in every integer value. To understand the fundamental physics that results in these anomalous transport properties, we analyze the Berry curvature density in the corresponding bulk system that reflects the nature of the QAHE.

For the disordered system, we use a generalized Berry curvature and Kubo formula to calculate conductances in real space instead of reciprocal space which can be expressed asBerryanalysis

In the above equations, $\Omega(\epsilon)=\frac{1}{S}{\rm Tr}[\hat{\Omega}\delta(\epsilon-H)]$ describes the Berry curvature density in the energy spectrum with $\hat{\Omega}$ being the Berry curvature operator $\hat{\Omega}_{\alpha}=\sum_{\alpha}|\alpha\rangle\langle\alpha|\$ where $|\alpha\rangle$ indicates the eigenvector of $|\hbar w_{\alpha}\rangle$ in the disorderd system. $S$ is the area of the two-dimensional system and $v_{x/y}$ is the velocity operator in the $x$ or $y$ direction. $f(\epsilon)$ is the Fermi-Dirac distribution.

In the bilayer graphene, the transport properties of nonmagnetic or spin-flip disorder doped systems are almost same as that in the square lattice with Chern number ${\mathcal{C}}=1$ and monolayer graphene with Chern number ${\mathcal{C}}=2$, which have been studied in our previous paper QAHE20 . Therefore, in this article, we mainly explore the mechanism of anomalous transport properties in the magnetic topological insulator thin films.

In the magnetic topological insulator thin films, Fig. 4 shows the evolution of the averaged Berry curvature density $\Omega$ and Hall conductance $\sigma_{xy}$ as a function of the Fermi energy $E$ for different spin-flip disorders strengths $W/B=0.00,0.60,1.10,1.60,1.85,2.15,2.40,2.95,3.45,3.65,3.90,$ $4.25,4.55$. At $W/B=0.00$ [see Fig. 4(a₁)], we can identify two Berry curvature peaks in either valence bands or conduction bands as $\rm P_{D1}^{v,c}$ that from the contribution of massive Dirac bands, and $\rm P_{R1}^{v,c}$ that from the contribution of remaining bands, where $\rm v/c$ denotes valence/conduction bands and $\rm D1/R1$ denotes the first Dirac/remaining bands. One can find that for Fermi energies inside the bulk band gap, $\Omega(E)=0$ and $\sigma_{xy}=2e^{2}/{h}$, and the Hall conductance preserve $\sigma_{xy}(E)=\sigma_{xy}(-E)$ due to the electron-hole symmetry of energy bands. In the presence of weak disorders [see Figs. 4(b)$-$4(d)], the two Berry curvature peaks $\rm P_{D1}^{v,c}$ move towards each other and shrink the band gap. The contribution of areas covered by red/green color to Hall conductance are about $0.5e^{2}/{h}$. Meanwhile, the two Berry curvature peaks $\rm P_{R1}^{v,c}$ are almost fixed and broaden. When the disorder strength approaches the first critical point $W/B\approx 1.85$ [Fig. 4(e)], the bulk gap begins closing and the Berry curvature peaks $\rm P_{R1}^{v,c}$ start to exchange with each other. If the disorder strength continues to increase [see Figs. 4(f)-(h)], one can find that the two peaks $\rm P_{R1}^{v,c}$ make a interchange with the bulk gap reopening, and the Hall conductance inside the bulk gap now becomes $\sigma_{xy}=e^{2}/{h}$, which is consistent with above transport calculations. At even stronger disorders [see Fig. 4(i)], one can see that the two peaks $\rm P_{R1}^{v,c}$ vanish and instead two new peaks appear, which can be identified as $\rm P_{D2}^{v,c}$ followed by the similar definitions above. When the disorder strength is strong [see Figs. 4(j)-(l)], the two Dirac peaks $\rm P_{D2}^{v,c}$ undergo the second interchange process, and finally the Hall conductance inside the bulk gap is zero, indicating the system becomes the Anderson insulator.

In contrast, Fig. 5 displays how the averaged Berry curvature density and Hall conductance evolve for the magnetic topological insulator thin films with nonmagnetic disorders. With the increase of disorder strength, although the two Dirac peaks move towards each other shrinking the bulk gap, the most noticeable difference from the case of spin-flip disorders is the lack of interchange of Berry curvature carried by two Dirac peaks. Therefore, the system with nonmagnetic disorders does not undergo a Berry-curvature mediated metallic phase but transit directly from the QAHE state to the Anderson insulating phase.

From above observed anomalous transport phenomenon in large Chern number system, we can now provide a phenomenological picture of topological charge to understand it.

In the magnetic topological insulator thin films, the continuous model of half Bernevig–Hughes–Zhang model is $H=k_{x}\sigma_{x}+k_{y}\sigma_{y}+m\sigma_{z}$. Owing to infinite domain of integration instead of first Brillouin zone in reciprocal space, the Hall conductance carried by a Dirac point is about $0.5e^{2}/h$. Before band inversion, the Hall conductance carried by Dirac point and remaining bands is respectively $0.5e^{2}/h$ and $-0.5e^{2}/h$ in valence bands, thus the total Hall conductance carried in valence bands is zero. After band inversion, the Hamiltonian is transformed to $H=k_{x}\sigma_{x}+k_{y}\sigma_{y}-m\sigma_{z}$. Now the Hall conductance carried by Dirac point and remaining bands is respectively $-0.5e^{2}/h$ in valence bands, so we could realize $-e^{2}/h$ QAHE. Here the disorder strength plays a role of a mass term in aforementioned band inversion mechanism. From the topological point of view, the quantized Hall conductances are analogous to four Skyrmions, as shown in Fig.  6(a), where valence/conduction bands carry four topological charges of +1 +1 -1 -1 labelled as “A”, “B”, “C”, “D”, and spins pointing outwards (inwards) is +1 (-1). When the spin-flip disorders are applied [see Fig. 6(b)], all Skyrmions will be divided to eight merons (labelled as ”A1”, ”A2”, “B1”, “B2”, “C1”, “C2”, “D1” and “D2”). When the disorder strength increasing to the first critical point [see Fig. 6(c)], merons B2 and C1 move towards each other and begin to interchange. At stronger disorder strength [see Fig. 6(d)], merons B2 and C1 have finished the interchange, thus the Hall conductance in valence bands is transit to the lower plateau $\sigma_{xy}=e^{2}/{h}$. Note that, merons D1 and A2 would not be influenced by the spin-flip disorders until disorder strength approaches the second critical point. When the disorder strength continuing to increase [see Fig 6(e)-(f)], merons D1 and A2 undergo the interchange process, thus the total Hall conductance vanishes.

In the bilayer graphene system, the mechanism of Berry curvature interchange is different from that in magnetic topological insulator thin films. The key point is that for all four Skyrmions are totally equivalent. Fig. 7 displays the evolution of topological charges carried by the spin-textures of the valence and conduction bands in bilayer graphene system. In the presence of weak disorder [see Fig. 7(a)-(b)], four Skyrmions (for simplicity, we draw four Skyrmions. Actually, there are eight Skyrmions in bilayer graphene because of the Chern number $\mathcal{C}=4$) are scattered to eight merons labelled as “A1”, “A2”, “B1”, “B2”, “C1”, “C2”, “D1” and “D2”. At stronger disorder [see Fig. 7(c)], the merons A2 and C1, B2 and D1 not make an interchange step by step, but interchange at the same time. After the interchange [see Fig. 7(d)], the net charge in valence band is zero, indicating the system enters the Anderson insulating phase.

As aforementioned discussion, we know that the mechanisms of the phase transition are different in the bilayer graphene and magnetic topological insulator thin films based QAHE system. The reason is that in the bilayer graphene system, the Skyrmions induced by nonzero Chern number are equivalent leading to the undistinguished feature during the phase transition, making the system enters the Anderson insulator after once interchange. Whereas in the magnetic topological insulator thin films, repeated band inversion will produce multiple unequal Skymions and the destruction condition of these Skymions are disparate by spin- flip disorders.