Chirality Breaking of Majorana Edge Modes Induced by Chemical Potential Shifts

Abstract

Quantum anomalous Hall insulator-superconductor heterostructures are predicted to host chiral Majorana fermions as edge modes, which is essential for topological quantum computing applications. Although the edge states have been extensively studied at zero chemical potential $μ= 0$, the practically relevant regime with a shifted chemical potential ($μ\neq 0$) remains less explored. Here, we present an analytical treatment of the edge states for $μ\neq 0$, deriving an approximate but highly accurate solution applicable to realistic experimental parameters. Surprisingly, we find that the energy dispersion of the edge band exhibits nonlinearity and transforms into a twisted, braid-like structure within specific parameter ranges. This unique braid-like band leads to non-chirality of the edge modes, allowing propagation in both directions.

Figures (4)

• Figure 1: (a) Illusatration of chiral Majorna fermion propagating at the edge of 2D material, and (b) linear energy dispersion of the edge states. The 2D material could be 2D $p$-wave superconductor ReadGreen2000, or heterostructure form by QAH insulator and $s$-wave supercondutor Qi_2010. (c) Illustration for the chirality-breaking of Majorana edge modes under chemical potentail shift. (d) The dispersion relation of edge states shows a twisted, barid-like shape so that each edge band intersect Fermi level three times.
• Figure 2: Edge state wave function under a chemical potential shift. Only the electron component is displayed, as the hole component is identical to the electron component. (a) $|\mu|=|m|$, where the spin-up component of the edge state $\Psi_{e\uparrow}$ decays exponentially, while the spin-down component $\Psi_{e\downarrow}$ exhibits an exponential decay modified by a linear factor $(1+\frac{2m}{A} y)$. (b) $|\mu|>|m|$, where the edge states oscillate and decay simultaneously.
• Figure 3: Energy spectra of the heterostructure system with various parameters. The edge bands are highlighted in red and blue. (a) $m=-0.6 \, \mathrm{meV}$, $\mu=1 \, \mathrm{meV}$. (b) $m=2 \, \mathrm{meV}$, $\mu=0.5 \, \mathrm{meV}$. (c) $m=-2 \, \mathrm{meV}$, $\mu=2 \, \mathrm{meV}$, where the edge bands form a braid-like structure. (d) $m=-2.5 \, \mathrm{meV}$, $\mu=2 \, \mathrm{meV}$. Other parameters are identical for (a--d): $\Delta_s = 1 \, \mathrm{meV}$,$A = 0.3 \, \mathrm{meV}\cdot\mu\mathrm{m}$, and $B = 1.5 \times 10^{-4} \, \mathrm{meV}\cdot\mu\mathrm{m}^2$, with a lattice constant $a = 5 \, \mathrm{nm}$ and a length $L_y = N_y a = 2 \, \mu\mathrm{m}$ along the $y$ direction.
• Figure 4: This figure illustrates the topological phase diagram as a function of $m$ and $\mu$, with the shaded regions representing areas where braid-like edge bands emerge. Panels (b)-(e) correspond to energy spectra at specific parameter sets labeled in (a), only the lowest band is shown for clearity. Orther parameter are setting the same as that in Fig. 3.