Disorder-induced topological transitions in a multilayer topological insulator

We examine the impact of nonmagnetic disorder on the electronic states of a multilayer structure comprising layers of both topological and conventional band insulators. Employing the Burkov-Balents model with renormalized tunneling parameters, we generate phase diagrams correlating with disorder, demonstrating that nonmagnetic disorder can induce transitions between distinct topological phases. The subsequent section of our investigation focuses on the scenario where disorder is unevenly distributed across layers, resulting in fluctuations of the interlayer tunneling parameter, termed off-diagonal disorder. Furthermore, we determine the density of states employing the self-consistent single-site diagram technique, expanding the Green function in relation to the interlayer tunneling parameter (locator method). Our findings reveal that off-diagonal disorder engenders delocalized bulk states within the band gap. The emergence of these states may lead to the breakdown of the anomalous quantum Hall effect (AQHE) phase, a phenomenon that has garnered significant attention from researchers in the realm of topological heterostructures. Nonetheless, our results affirm the stability of the Weyl semimetal phase even under substantial off-diagonal disorder.

Figure 1

Schematic representation of the system under study. Horizontal arrows on the right panel represent chiral edge modes in the TI films. Different colors of arrows (green and red) correspond to different chiralities. A unit cell of height $d$ contains one TI layer and one normal insulator layer.

Figure 2

Diagram series for the self-energy part (5) in the single-impurity cross technique in the T-scattering matrix approximation.

Figure 3

Schematic representation of the dependence of the spatial distribution of edge modes on the topological mass value. A smaller value of $|\overline{m}|$ corresponds to a greater depth penetration.

Figure 4

Dependence of the value ${\overline{\mathrm{\Delta }}}_S$ on $U_0$. The upper panel shows the case where $m<0$ and $t<\gamma$ and an increase in $U_0$ leads to an increase in ${\overline{\mathrm{\Delta }}}_S$. It can be seen that ${\overline{\mathrm{\Delta }}}_S={\mathrm{\Delta }}_D$ for some value of $U_0$. This leads to a transition from the topological phase to the normal. This transition can be understood if we use the formula for the $Z_2$ topological invariant from the work [42], rewritten as ${(-1)}^{\nu }=\text{sgn}({\overline{\mathrm{\Delta }}}_S-{\mathrm{\Delta }}_D)$. The lower panel demonstrates the case when films in a multilayer structure are Anderson insulators with $m>0$ and $t>\gamma$, and an increase in the value of $U_0$ leads to an increase in the topological mass and a decrease in ${\overline{\mathrm{\Delta }}}_S$.

Figure 5

Phase diagrams for the parameters ${\mathrm{\Delta }}_S$ and $U_0$. The upper panel shows the situation ${\mathrm{\Delta }}_Z<{\mathrm{\Delta }}_D$, when only the phases of an normal insulator and a Weyl semimetal can be realized. The lower panel demonstrates the situation when ${\mathrm{\Delta }}_Z>{\mathrm{\Delta }}_D$, when in addition to the indicated phases, the implementation of the AQHE phase is possible. From the figures it is clear that disorder can be the cause of transitions between the mentioned phases.

Figure 6

Spectrum of Hamiltonian (12) for different parameter values: (a) ${\mathrm{\Delta }}_Z=0$, an ordinary insulator, (b) $|{\mathrm{\Delta }}_Z|<|{\overline{\mathrm{\Delta }}}_S-{\mathrm{\Delta }}_D|$, ordinary insulator, (c) $|{\overline{\mathrm{\Delta }}}_S-{\mathrm{\Delta }}_D|<|{\mathrm{\Delta }}_Z|<|{\overline{\mathrm{\Delta }}}_S+{\mathrm{\Delta }}_D|$, Weyl semimetal phase, and (d) $|{\mathrm{\Delta }}_Z|>|{\overline{\mathrm{\Delta }}}_S+{\mathrm{\Delta }}_D|$, inverted spectrum-AQHE state.

Figure 7

Schematic representation of off-diagonal disorder: tunnel parameter ${\mathrm{\Delta }}_S$ as a random function of layer number.

Figure 8

Density of states as a function of energy for two phases in the absence of Zeeman splitting (${\mathrm{\Delta }}_Z=0$). The upper panel demonstrates normal insulator phase with ${\mathrm{\Delta }}_S=25\text{meV}$ and ${\mathrm{\Delta }}_D=20\text{meV}$. The band gap regions corresponding to different density of states curves are shaded in different colors. An increase in ${\eta }_0$ leads to a decrease in the band gap and the appearance of delocalized states. The lower panel corresponds to topological insulator phase with ${\mathrm{\Delta }}_S=20\text{meV}$ and ${\mathrm{\Delta }}_D=25\text{meV}$. An increase in ${\eta }_0$ first leads to an increase in the band gap (to approximately ${\eta }_0=13\text{meV}$), and then the band gap quickly collapses, also leading to delocalized states. Thus, the topological phase is stable at small values of ${\eta }_0$. Only positive energies are shown, because the curves are symmetrical about the vertical axis. In both cases the panels include insets demonstrating the dependence of the band gap $\tilde{\mathrm{\Delta }}$ on ${\eta }_0$.

Figure 9

Density of states as a function of energy in the presence of Zeeman splitting (${\mathrm{\Delta }}_Z\ne 0$) at ${\mathrm{\Delta }}_S=25\text{meV}$ and ${\mathrm{\Delta }}_D=20\text{meV}$. The upper panel demonstrates Weyl semimetal phase ($|{\overline{\mathrm{\Delta }}}_S-{\mathrm{\Delta }}_D|<{\mathrm{\Delta }}_Z=20\text{meV}<|{\overline{\mathrm{\Delta }}}_S+{\mathrm{\Delta }}_D|$). The density of states changes very little with increasing ${\eta }_0$. Thus, the Weyl point is stable to nondiagonal disorder. The lower panel shows AQHE phase (${\mathrm{\Delta }}_Z=50\text{meV}>|{\overline{\mathrm{\Delta }}}_S+{\mathrm{\Delta }}_D|$). It can be seen that as ${\eta }_0$ increases, the band gap only increases, which indicates the stability of the AQHE phase relative to off-diagonal disorder. In the case of AQHE phase, we add the inset where the dependence of the band gap $\tilde{\mathrm{\Delta }}$ on ${\eta }_0$ is presented.

Figure 10

Diagrammatic expansions for the quantities $\langle \sigma \rangle$ and $\overline{t}$. Gray circles in the lower diagrams indicate identical indices. Intermediate indices can be any, but differ from the edge indices (gray). This requirement simply corresponds to the exclusion of repeated counting of the same charts.