Braiding rule of boundary Majorana-like zero mode

TL;DR

This paper uncovers the braiding rules of boundary Majorana-like zero modes in vortex-texture Kekule graphene, demonstrating their topological protection, tunability, and potential for implementation in acoustic crystals, advancing topological quantum state control.

Contribution

It reveals the braiding rule of boundary MLZMs in Kekule graphene and proposes an acoustic crystal implementation, expanding the scope of topological state manipulation.

Findings

Boundary MLZMs are protected by the Zak phase.

Multiple MLZMs can be constructed and braided.

Implementation scheme in acoustic crystals is provided.

Abstract

The study of topological states has become an important topic in both solid-state systems and artificial structures such as photonic crystals and phononic crystals. Among them, Majorana zero modes, which exhibit nontrivial braiding process, have attracted extensive research interest. The analog of Majorana zero modes in classical waves, or the Majorana-like zero modes (MLZMs), have also got a lot of attention recently. However, the vast majority of previous works concerned with MLZMs that were bounded to vortexes inside the bulk. Here in this work, we unveil the braiding rule of MLZMs that are tunable around the boundary of a vortex-texture Kekule modulated graphene. We show that the existence of these zero-dimensional boundary MLZMs is protected by the Zak phase of 1D boundary states. As such, we are able to construct multiple MLZMs and analyze the corresponding braiding process. In addition, we also provide an implementation scheme of the boundary MLZMs in acoustic crystals. The tunability of the boundary MLZMs proposed herein offer new freedom for topological states braiding in both solid-state systems and artificial structures.