Majorana Stripe Order at the Surface of a 3D topological Insulator

The effect of interactions in topological states is a topical issue; not only interaction-enabled topological phases but also novel symmetry-breaking phases and phase transitions are possible. Here we study the effect of interactions on Majorana zero modes (MZMs) bound to a square lattice of vortices in 2D topological superconductors. In the special neutrality condition, where the usual hybridization term for MZMs is prohibited by symmetry, we show that a minimal model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo simulation. Guided by an insight from a further duality mapping to a compass model, we demonstrate that the interactions induce a Majorana stripe order spontaneously breaking translational and rotational symmetries. Away from the neutrality condition, mean field theory suggests a quantum critical point induced by hybridization, beyond which a Dirac cone appears in the excitation spectrum.