Unitary symmetry-protected non-Abelian statistics of Majorana modes

Jian-Song Hong, Ting-Fung Jeffrey Poon, Long Zhang, and Xiong-Jun Liu

Phys. Rev. B 105, 024503 – Published 5 January 2022

Abstract

Symmetry-protected topological superconductors (TSCs) can host multiple Majorana zero modes (MZMs) at their edges or vortex cores, while whether the Majorana braiding in such systems is non-Abelian in general remains an open question. Here, we uncover in theory the unitary symmetry-protected non-Abelian statistics of MZMs and propose the experimental realization. We show that braiding two vortices with each hosting $N$ MZMs protected by commutative or noncommutative unitary symmetries generically reduces to $N$ independent sectors, with each sector braiding two different Majorana modes. This renders the unitary symmetry-protected non-Abelian statistics. We demonstrate the proposed non-Abelian statistics in a spin-triplet TSC which hosts two MZMs at each vortex and, interestingly, can be precisely mapped to a quantum anomalous Hall insulator. Thus the unitary symmetry-protected non-Abelian statistics can be verified in the latter insulating phase, with the application to realizing various topological quantum gates being studied, and an experimental scheme of observation being proposed. This work suggests that there are a much broader range of symmetry-protected topological superconductors which host non-Abelian Majorana modes.

Received 3 November 2020
Revised 27 October 2021
Accepted 15 December 2021

DOI:https://doi.org/10.1103/PhysRevB.105.024503

Physics Subject Headings (PhySH)

Cold and ultracold molecules Majorana bound states Spin-orbit coupling Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jian-Song Hong^1,2,*, Ting-Fung Jeffrey Poon^1,2,*, Long Zhang ^1,2, and Xiong-Jun Liu ^1,2,3,4,†

¹International Center for Quantum Materials and School of Physics, Peking University, Beijing 100871, China
²Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
³CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
⁴Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China

^*These authors contributed equally to this work.
^†Author to whom correspondence should be addressed: xiongjunliu@pku.edu.cn

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Vol. 105, Iss. 2 — 1 January 2022

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Images

Figure 1
Schematic diagrams for MZMs residing at vortices in a unitary symmetry-protected TSC. The gray circles with arrows denote vortices, the blue dots are MZMs, and the dashed lines represent the branch cuts of MZMs. (a1) and (a2) The braiding of multiple pairs of MZMs is reduced into multiple individual sectors, each of which braids two MZMs independently. (b) Sketch of $B_{23}$ in the concrete model. The local MZMs $γ_{i a}$ and $γ_{i b}$ define a complex fermion $η_{i}$ . The nonlocal complex fermions $f_{i a (b)}$ are defined by MZMs in different vortices.
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Figure 2
Numerical results of braiding two vortices with perturbations. (a) Evolution of MZM wave function in a full braiding. (b) and (c) Evolution with symmetry-preserving randomized chemical potential $δ μ$ and disordered spin-orbit coupling $δ t_{SO}$ . (d) Evolution with symmetry-breaking $s$ -wave pairing $Δ_{s}$ . The non-Abelian braiding is confirmed in (a)–(c), in that $η (t) = 〈 γ_{1 a} (0) | γ_{1 a} (t) 〉$ (denoted by blue curves) is equal to $- 1$ at $t = 2 T$ (similar for other MZMs), and breaks down in (d). The adiabatic condition is satisfied in that $ζ (t) = \sum_{j = 1, 2} [| 〈 γ_{1 a} (t) | γ_{j a} (0) 〉 |^{2} + 〈 γ_{1 a} (t) | γ_{j b} (0) 〉 |^{2}]$ (denoted by red curves) returns to unity after a single ( $t = T$ ) and a full ( $t = 2 T$ ) braiding. Here, $μ = 1 / m$ and $Δ = 1 / 2 m$ .
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Figure 3
Numerical simulation of a full braiding between $η_{2}$ and $η_{3}$ . (a) Orange balls denote the lattice sites, and vortices (antivortices) are denoted by blue (red) circles. (b) The evolution of wave function $| η_{1 -} (t) 〉$ . The system evolves into a superposition of the four single-particle states while braiding $η_{2}$ and $η_{3}$ once. After a full braiding, $| η_{1 -} (t) 〉$ evolves into $| η_{1 +} 〉$ . (c) The energy $E$ of $| η_{1 -} (t) 〉$ as a function of time. The absolute value of $| η_{1 -} (0) 〉$ is denoted by $E_{0}$ . The dashed curves show the energy evolution with certain randomization $δ m_{z}$ in the $m_{z}$ term. After a full braiding, all three curves arrive at the positive energy $E_{0}$ , demonstrating the robustness of non-Abelian statistics. Here, we set $m_{z} = 2 t_{0}$ and $t_{SO} = t_{0}$ .
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Figure 4
(a) Sketch of the experimental realization scheme. The atoms reside on a primary checkerboard lattice (black lines) with a staggered on-site Zeeman potential $V_{s}$ . The Raman potential $M$ induces the spin-flip nearest-neighbor hopping. A shaking lattice $V_{t}$ is applied to drive the spin-conserved hopping by compensating the neighboring on-site energy difference. (b) The phase of Raman potential $θ_{M}$ with $l = 1$ . (c) The phase of the spin-flip hopping $θ$ in the $x$ direction produced by the Raman potential with $l = 1$ . (d) The energy spectrum of the system with a vortex ( $l = 1$ ) and an antivortex ( $l = - 1$ ). See details in Ref. [63].
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