Chiral and counter-propagating Majorana fermions in a p-wave superconductor

Haiping Hu,Erhai Zhao,Indubala I Satija

doi:10.1088/1367-2630/ab5cad

Haiping Hu, Erhai Zhao + Show 1 more

Open Access

https://doi.org/10.1088/1367-2630/ab5cad

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Abstract

Chiral and helical Majorana fermions are two archetypal edge excitations in two-dimensional topological superconductors. They emerge from systems of different Altland–Zirnbauer symmetries and characterized by and topological invariants respectively. It seems improbable to tune a pair of co-propagating chiral edge modes to counter-propagate in a single system without symmetry breaking. Here, we explore the peculiar behaviors of Majorana edge modes in topological superconductors with an additional ‘mirror’ symmetry which changes the bulk topological invariant to type. A theoretical toy model describing the proximity structure of a Chern insulator and a px-wave superconductor is proposed and solved analytically to illustrate a direct transition between two topologically nontrivial phases. The weak pairing phase has two chiral Majorana edge modes, while the strong pairing phase is characterized by mirror-graded Chern number and hosts a pair of counter-propagating Majorana fermions protected by the mirror symmetry. The edge theory is worked out in detail, and implications to braiding of Majorana fermions are discussed.

Highlights

A defining feature of topological quantum matter [1, 2] is the existence of protected boundary/edge modes due to the nontrivial topology of the bulk material
The weak pairing phase has two chiral Majorana edge modes, while the strong pairing phase is characterized by O-graded Chern number and hosts a pair of counter-propagating Majorana fermions
Gapped bulk Hamiltonians can be classified into different Altland-Zirnbauer (AZ) classes based on their symmetries [3,4,5,6]

Summary

Introduction

A defining feature of topological quantum matter [1, 2] is the existence of protected boundary/edge modes due to the nontrivial topology of the bulk material. We seek a tunable 2D superconductor that changes from the host of a pair of chiral Majorana modes to the host of a pair of counter-propagating Majorana modes, schematically shown in Fig. 1(a) and (b), as the magnitude of the superconducting pairing is increased At first sight, such a direct Z to Z2 transition seems impossible without any symmetry breaking because these two types of edge modes require distinct symmetries of the bulk Hamiltonian according to the standard AZ classification [3,4,5,6]. As we show explicitly below, the presence of additional symmetry (besides time reversal, particle-hole and chiral symmetry) gives rise to richer physics beyond the AZ classes, and it is possible to realize both types of edge modes within the same material This implies that the direction of the Majorana fermions can be controlled to yield new quantum gates.

Chern number defined as

Chern number