Quasi-one-dimensional quantum anomalous Hall systems as new platforms for scalable topological quantum computation

Quantum anomalous Hall insulator/superconductor heterostructures emerged as a competitive platform to realize topological superconductors with chiral Majorana edge states as shown in recent experiments [He et al. Science 357, 294 (2017)]. However, chiral Majorana modes, being extended, cannot be used for topological quantum computation. In this work, we show that quasi-one-dimensional quantum anomalous Hall structures exhibit a large topological regime (much larger than the two-dimensional case) which supports localized Majorana zero energy modes. The non-Abelian properties of a cross-shaped quantum anomalous Hall junction is shown explicitly by time-dependent calculations. We believe that the proposed quasi-one-dimensional quantum anomalous Hall structures can be easily fabricated for scalable topological quantum computation.

Figure 1

(a) Schematic plot of a QAH insulator with two chiral edge modes (red arrows) in proximity to an $s$-wave superconductor (SC). (b) The QAH insulator is narrowed down such that the two chiral edge modes are coupled to form a single helical conducting channel (red dashed arrows); inducing superconductivity gives rise to two Majorana modes ${\gamma }_1$ and ${\gamma }_2$. (c) The energy spectrum of quasi-1D QAH insulator with width $L_y=800\mathrm{nm}$. The two edge modes on opposite sides are approximately decoupled as the $L_y$ is much larger than the localization length of the chiral edge modes. (d) The energy spectrum with $L_y=100\mathrm{nm}$. The dashed line indicates the possible location of the chemical potential in the topological regime. A gap will open at the chemical potential upon proximity coupling to a superconductor. A tight-binding version of $H_{\mathrm{QAH}}$ is used for (c) and (d) and the parameters are $\hslash v_F=3\mathrm{eV}\AA$, $m_0=-5\mathrm{meV}$, $m_1=15\mathrm{eV}{\AA }^2$, $M_z=8\mathrm{meV}$.

Figure 2

(a) The number $n$ denotes the number of bands of the quasi-1D QAH (without superconductivity) cut by the Fermi energy as a function of chemical potential $\mu$ and effective Zeeman field $M_z$. The width and parameters of the quasi-1D system are the same as in Fig. 1. (b) The topological invariant $N_{\text{BDI}}$ in the superconducting phase as a function of $\mu$ and $M_z$. The parameters are the same as (a) except that a pairing potential $\mathrm{\Delta }=1\mathrm{meV}$ is introduced on the top surface.

Figure 3

(a) Schematic plot of two-terminal device for the detection of Majorana zero modes. A lead is attached to a QAH system while the narrow region of the QAH system is in proximity to an $s$-wave superconductor to create the Majorana zero energy mode. An electron mode from the edge state of the QAH system (black arrows) is reflected as a hole mode (blue arrows) by a Majorana mode $\gamma$ (red dot). (b) The zero-bias conductance $G$ from the lead to the superconductor as a function of Zeeman field $M_z$ with chemical potential $\mu =5\mathrm{meV}$. (c) The two terminal conductance vs electric voltage eV for different $\mu$ with $M_z=6\mathrm{meV}$ and $\mathrm{\Delta }=1\mathrm{meV}$. Other parameters are the same as those in Fig. 2.

Figure 4

(a) Schematic plot of a cross-shaped QAH system in proximity to a superconductor with four tunable gates (G1–G4) on the top. Initially, when G1 and G3 are turned on with G2 and G4 are turned off, the system is divided into three topologically nontrivial parts with six Majorana modes (${\gamma }_1\text{–}{\gamma }_6$). (b) The projection of wave function $\varphi \left(t\right)$ onto the initial eigenstate $|n_1{n}_2\rangle$ as a function of time, where $\varphi \left(t\right)=\mathbb{T}{e}^{i\int dt{H}_C/\hslash }|n_1{n}_2\rangle$ with the time-ordered operator $\mathbb{T}$ and $H_C$, is a realistic tight-binding model describing the cross-shaped QAH junction. The six-stage process of braiding ${\gamma }_2$ and ${\gamma }_3$ is described in the main text. The final states are orthogonal to the initial states after braiding and non-Abelian statistics is clearly demonstrated. The total braiding time for the six steps is $6T=14400\hslash /\mathrm{\Delta }\approx 9.5\mathrm{ns}$. The details of the calculations can be found in the Supplemental Material [44].