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Chiral and helical Majorana edge modes are two archetypal gapless excitationsof two-dimensional topological superconductors. They belong to superconductorsfrom two different Altland-Zirnbauer symmetry classes characterized by$\mathbb{Z}$ and $\mathbb{Z}_2$ topological invariant respectively. It seemsimprobable to tune a pair of co-propagating chiral edge modes tocounter-propagate without symmetry breaking. Here we show that such a directtopological transition is in fact possible, provided the system possesses anadditional symmetry $\mathcal{O}$ which changes the bulk topological invariantto $\mathbb{Z}\oplus \mathbb{Z}$ type. A simple model describing the proximitystructure of a Chern insulator and a $p_x$-wave superconductor is proposed andsolved analytically to illustrate the transition between two topologicallynontrivial phases. The weak pairing phase has two chiral Majorana edge modes,while the strong pairing phase is characterized by $\mathcal{O}$-graded Chernnumber and hosts a pair of counter-propagating Majorana fermions. The bulktopological invariants and edge theory are worked out in detail. Implicationsof these results to topological quantum computing based on Majorana fermionsare discussed.

Chiral to helical Majorana fermion transition in a $p$-wave superconductor