Quantum phase transitions of a disordered antiferromagnetic topological insulator

We study the effect of electrostatic disorder on the conductivity of a three-dimensional antiferromagnetic insulator (a stack of quantum anomalous Hall layers with staggered magnetization). The phase diagram contains regions where the increase of disorder first causes the appearance of surface conduction (via a topological phase transition), followed by the appearance of bulk conduction (via a metal-insulator transition). The conducting surface states are stabilized by an effective time-reversal symmetry that is broken locally by the disorder but restored on long length scales. A simple self-consistent Born approximation reliably locates the boundaries of this so-called “statistical” topological phase.

Figure 1

Stack of antiferromagnetically ordered layers. Each layer is insulating in the interior but supports a chiral edge state (arrows) because of the quantum anomalous Hall effect. Interlayer hopping (in the $z$ direction) produces an anisotropic Dirac cone of surface states on surfaces perpendicular to the layers. The unpaired Dirac cone is robust against disorder, as in a (strong) topological insulator, although time-reversal symmetry is broken locally.

Figure 2

Energy spectrum of the AFTI Hamiltonian Eq. (2.6), with $t_z=0.4$, for a stack of 16 layers in the $z$ direction with periodic boundary conditions. The layers are infinitely wide in the $x$ direction and truncated at 16 lattice sites in the $y$ direction. At $\mu =\pm 1$ the system is in the AFTI phase, with a nondegenerate Dirac cone of surface states centered at the edge of the Brillouin zone ($-2<\mu <0$) or at the center of the Brillouin zone ($0<\mu <2$). At $\mu =0$ the bulk gap closes at a pair of twofold degenerate Weyl cones, one at the center and one at the edge of the Brillouin zone. In this plot a finite gap remains for $\mu =0$, because of the confinement in the $y$ direction.

Figure 3

Conductance at the Weyl point for periodic boundary conditions, according to Eq. (2.20) (solid curve) and the asymptotic form for large aspect ratio Eq. (2.22) (dashed). The data points are calculated from the AFTI Hamiltonian Eq. (2.6), at $\mu =2$, $t_z=0.4$, for a lattice of eight layers in the $z$ direction, with periodic boundary conditions in the $x$ and $y$ directions (red dots) and for hard-wall boundary conditions (black crosses).

Figure 4

Same as described in the caption of Fig. 3, but for the Fano factor at the Weyl point.

Figure 5

Color-scale plot of the conductance of a disordered AFTI, calculated numerically from the Hamiltonian Eq. (2.6) for current flowing perpendicular to a stack of 20 layers. Each layer has dimensions $20\times 20$ with periodic boundary conditions, the interlayer coupling is $t_z=0.4$. The topological insulator phase (AFTI), the trivial insulator phase (I), and the metallic phase (M) are indicated in the plot. The white curves are the phase boundaries resulting from the self-consistent Born approximation (SCBA). The Anderson transition between a metal and a trivial insulator is not captured by the SCBA.

Figure 6

Disorder averaged conductance for three system sizes. Panels (a) and (b) show the transition between a metal (M) and an insulator which is topologically trivial (I) or nontrivial (AFTI). Panel (c) show the trivial-to-nontrivial insulator transition. The scale-independent conductance at the critical point of the phase transition is indicated by an arrow. The curves are guides to the eye. Data points from panels (a) and (b) are averages of over 20 000 disorder configurations, data points from panel (c) are averages of over 200 configurations.

Figure 7

Disorder averaged conductance on the line $\mu =0$, within the AFTI phase for weak disorder and metallic for strong disorder. The ballistic conductance at the Weyl point is indicated.