Stability of Majorana Edge Zero Modes against Interactions

We study an interacting Majorana chain with an open boundary condition. In the case without interactions, the system shows a prototypical Majorana edge zero mode in the sector of the ground state with a spectral gap above the sector. We prove that both of the Majorana edge zero mode and the non-vanishing spectral gap are stable against generic weak interactions whose fermion parity is even. We also deal with the corresponding ladder systems, and discuss the $\mathbb{Z}_2$ index for the Majorana edge zero modes.