Unified model for non-Abelian braiding of Majorana and Dirac fermion zero modes

Abstract

Majorana zero modes (MZMs) are the most intensively studied non-Abelian anyons. The Dirac fermion zero modes in topological insulators, which are symmetry-protected doubling of MZMs under fermion number conservation, offer an alternative approach to explore non-Abelian anyons. However, a unified model that elucidates the braiding statistics of these types of topological zero modes remains absent. We show that the minimal Kitaev chain model beyond fine-tuning regime provides a unified characterization of the non-Abelian statistics of both MZMs and Dirac fermion zero modes in different parameter regimes. In particular, we introduce a minimal tri-junction setting based on the minimal Kitaev chain model and show it facilitates the unified scheme of braiding Dirac fermion zero modes, as well as the MZMs in the assistance of a Dirac mode. This unified minimal model provides deeper insights into non-Abelian statistics, demonstrating that the non-Abelian braiding of MZMs can be continuously extended to encompass Dirac fermion zero modes. The minimal Kitaev chain has been realized in coupled quantum dots [Nature 614, 445 (2023)]. Our extension, which demonstrates novel nontrivial phases with non-Abelian MZM pairs and Dirac zero modes emerging in the broader parameter regimes without fine-tuning, expands the accessible experimental parameter space and enhances the feasibility of observing non-Abelian statistics in the minimal Kitaev chain model.

Figures (3)

• Figure 1: (a) Sketch of the Y-junction jointed by minimal Kitaev chains. (b) The braiding operation corresponds to a closed loop (started and ended at the point marked in red) in the parameter space.
• Figure 2: (a) Schematic representation of the Majorana Hamiltonian [Eq. (\ref{['Hamiltonian_CaseIandII']})] under the conditions that $|\chi_j|=|\Omega_j|$. The Fock space we consider is spanned by six MZMs inside the dashed box. (b), (c) The energy spectra during the braiding as (b) Case I in TABLE \ref{['table']} that $|\chi_j|=|\Omega_j|$ and $\varphi_2,\varphi_3=0$. All the states in (b) are doubly degenerate; (c) Case II in TABLE \ref{['table']} that $|\chi_j|=|\Omega_j|$ and $\varphi_2,\varphi_3\ne0$ ($\varphi_2 = \pi/6$, and $\varphi_3= \pi/12$ here for illustration). $|\chi| \equiv \sqrt{|\chi_1|^2 + |\chi_2|^2 + |\chi_3|^2}$ is chosen as the energy unit.
• Figure 3: (a), (b) Schematic representation of the Majorana Hamiltonian [Eq. (\ref{['Hamiltonian_CaseIII']})] under the conditions of (a) Case III in TABLE \ref{['table']}; and (b) Case IV in TABLE \ref{['table']}. (c), (d) The energy spectra during the braiding as (c) Case III that $|\chi_j| \ne |\Omega_j|$ and $\varphi_2,\varphi_3=0$ ($|\Omega_i|/|\chi_i|=0.8$ here for illustration); and (d) Case IV that $|\chi_j| \ne |\Omega_j|$ and $\varphi_2,\varphi_3 \ne 0$ ($|\Omega_i|/|\chi_i|=0.8$, $\varphi_2 = \pi/6$, and $\varphi_3= \pi/12$ here for illustration). The energy unit $|\chi| \equiv \sqrt{|\chi_1|^2 + |\chi_2|^2 + |\chi_3|^2}$, and all the states in (c) and (d) are doubly degenerate.