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

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 2  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   Å type. A theoretical toy model describing the proximity structure of a Chern insulator and a p x -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.

Tuning the topology of $p$-wave superconductivity in an analytically solvable two-band model

“…Let V be the unitary transformation that diagonalizes O, V † OV = −τ z . In the eigenbasis of O, the Hamiltonian is decomposed into the direct sum 18 ,…”

Dispersive 1D Majorana modes with emergent supersymmetry in 1D proximitized superconductors via spatially-modulated potentials and magnetic fields

“…Moreover, chiral and helical 1D Majorana modes have been predicted to emerge as topologically protected edge states in the topologically nontrivial phase of a 2D superconductor or planar superconducting heterostructures. [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] In chiral topological superconductors, each edge carries one mode with a definite chirality, with modes at opposite edges having opposite spins and propagating in opposite directions. In helical topological superconductors instead, each edge carries a helical pair of counterpropagating modes having opposite spins.…”

“…In any case, 0D and 1D (chiral or helical) Majorana modes correspond to the boundary excitations of a topologically nontrivial bulk. [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] Despite the fact that theoretical models are rather simple, the experimental quest for signatures of Majorana quasiparticles has been challenging 6,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]40,45,46 since identical signatures can be explained without invoking Majorana quasiparticles in the presence of inhomogeneous potentials and disorder. [47][48][49][50][51][52][53][54]…”

1D Majorana Goldstinos and partial supersymmetry breaking in quantum wires

“…Generally, Majorana quasiparticles are topologically protected (d−1)-dimensional boundary excitations of a topologically nontrivial d-dimensional bulk. Specifically, 0-dimensional (0D) Majorana modes [3][4][5] correspond to the end states of 1D quantum systems with proximitized superconductivity, whereas chiral and helical 1D Majorana modes correspond to the edge states of 2D unconventional superconductors or planar superconducting heterostructures [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] respectively with broken or unbroken time-reversal symmetry.…”