Quantum phase transition of chiral Majorana fermions in the presence of disorder

We study the quantum phase transitions of a disordered two-dimensional quantum anomalous Hall insulator with $s$-wave superconducting proximity, which are governed by the percolation theory of chiral Majorana fermions. Based on symmetry arguments and a renormalization-group analysis, we show there are generically two phase transitions from Bogoliubov–de Gennes Chern number $\mathcal{N}=0$ to $\mathcal{N}=1(p+ip$ chiral topological superconductor) and then to $\mathcal{N}=2$, in agreement with the conclusion from the band theory without disorders. Further, we discuss the critical scaling behavior of the $e^2/2h$ conductance half plateau induced by the $\mathcal{N}=1$ chiral topological superconductor recently observed in the experiment. In particular, we compare the critical behavior of the half plateau induced by the topological superconductor with that predicted recently by alternative explanations of the half plateau and show that they can be distinguished in experiments.

Figure 1

(a) Illustration of the QAHI/TSC/QAHI junction, where a QAHI thin film is in proximity to an $s$-wave SC on top in the middle region. (b)–(d) Evolution of BdG Chern number $\mathcal{N}=0,1,2$ regions during the domain-flipping process, where each of the two chiral Majorana states ${\gamma }_a$ and ${\gamma }_b$ undergoes a delocalization transition. (b), (c), and (d) show configurations where $\mathcal{N}=2,1$, and 0 regions are extended, respectively.

Figure 2

(a) Formulation of the system into a $D$-class network model. (b) Illustration of the scattering of Majorana state ${\gamma }_b$ at network vertices.

Figure 3

(a) The propagation amplitude ${\mathcal{M}}_{bb}^{tot,n}$ of ${\gamma }^b$ near its delocalization transition, which has two peaks at ${\varphi }_2=0$ and $\pi$. (b) Illustration of the RG flow of ${\theta }_a,{\theta }_b$, and ${\varphi }_i$ near the delocalization transition of ${\gamma }^b$.

Figure 4

(a) The physical picture of a phase transition from $e^2/2h$ to $e^2/h$ after RG flow. (b) The picture of transition from zero to $e^2/2h$ after RG flow. (c) Predicted temperature dependence and critical behavior of ${\sigma }_{12}$ near the half plateau.