  Nodal topological superconductors display zero-energy Majorana flat bands at
generic edges. The flatness of these edge bands, which is protected by
time-reversal and translation symmetry, gives rise to an extensive ground-state
degeneracy. Therefore, even arbitrarily weak interactions lead to an
instability of the flat-band edge states towards time-reversal and
translation-symmetry-broken phases, which lift the ground-state degeneracy. We
examine the instabilities of the flat-band edge states of d_{xy}-wave
superconductors by performing a mean-field analysis in the Majorana basis of
the edge states. The leading instabilities are Majorana mass terms, which
correspond to coherent superpositions of particle-particle and particle-hole
channels in the fermionic language. We find that attractive interactions induce
three different mass terms. One is a coherent superposition of imaginary s-wave
pairing and current order, and another combines a charge-density-wave and
finite-momentum singlet pairing. Repulsive interactions, on the other hand,
lead to ferromagnetism together with spin-triplet pairing at the edge. Our
quantum Monte Carlo simulations confirm these findings and demonstrate that
these instabilities occur even in the presence of strong quantum fluctuations.
We discuss the implications of our results for experiments on cuprate
high-temperature superconductors.
