Observation of the Quantum Anomalous Hall Insulator to Anderson Insulator Quantum Phase Transition and its Scaling Behavior

Fundamental insight into the nature of the quantum phase transition from a superconductor to an insulator in two dimensions, or from one plateau to the next or to an insulator in the quantum Hall effect, has been revealed through the study of its scaling behavior. Here, we report on the experimental observation of a quantum phase transition from a quantum-anomalous-Hall insulator to an Anderson insulator in a magnetic topological insulator by tuning the chemical potential. Our experiment demonstrates the existence of scaling behavior from which we extract the critical exponent for this quantum phase transition. We expect that our work will motivate much further investigation of many properties of quantum phase transition in this new context.

Figure 1

Chemical potential dependence of topological quantum phase transition in magnetic topological insulator films. (a) A schematic diagram depicting the QAH to Anderson insulator quantum phase transition. For a smoothly varying disorder, three regions can be identified as a function of the chemical potential: the QAH phase; the metal phase; and the mixed region with coexisting QAH and metal phases. Further addition of short range disorder turns the metallic regions into an Anderson insulator. (b),(c) magnetic field ${\mu }_{0}H$ dependence of longitudinal resistance ${\rho }_{xx}$ and Hall resistance ${\rho }_{yx}$ at gate bias $V_g=+120\mathrm{V}$ (b) and $V_g=-200\mathrm{V}$ (c).

Figure 2

Gate-induced topological quantum phase transition from the QAH insulator to Anderson insulator. (a) Temperature dependence of the longitudinal resistance ${\rho }_{xx}$ for different $V_g$s. Inset: temperature dependence of the longitudinal conductance ${\sigma }_{xx}$ for different $V_g$s. (b),(c) Gate dependence of the longitudinal resistance ${\rho }_{xx}$ (b) and Hall resistance ${\rho }_{yx}$ (c) measured at different fixed temperatures. Inset of (b) shows hysteretic behavior upon sweeping $V_g$ at different temperatures.

Figure 3

Scaling analysis for the QAH insulator to Anderson insulator transition point. (a) Experimental data close to the QPT critical point collapse onto the same curve with good accuracy, in agreement with the one-parameter scaling law. The inset shows a zoom-in plot of the regime close to the QPT. (b) The critical exponent $\kappa$ and the critical gate voltage $V_g^c$ (inset), as well as their errors, have been obtained by using a well-developed statistical procedure (see Ref. [22]). The length of fitting curves (dashed lines) here also indicates the range of the experimental data used in the analysis.

Figure 4

Example of the numerical finite-size scaling analyses based on a generic QAH model (see Ref. [22] for details). The inset shows the normalized localization length $\mathrm{\Lambda }$ as a function of energy $E$, where the critical energy $E_c$ for the transition between an Anderson insulator ($E<E_c$) and a QAH insulator ($E>E_c$) is indicated. The normalized localization length $\mathrm{\Lambda }$ close to the QPT critical point obeys the one-parameter scaling law, as indicated by the fitting curves (dashed lines) in the main figure.