  We study the low-energy physics of a chain of Majorana fermions in the
presence of interaction and disorder, emphasizing the difference between
Majoranas and conventional (complex) fermions. While in the non-interacting
limit both models are equivalent (in particular, belong to the same symmetry
class BDI and flow towards the same infinite-randomness critical fixed point),
their behavior differs drastically once interaction is added. Our
density-matrix renormalization group calculations show that the complex-fermion
chain remains at the non-interacting fixed point. On the other hand, the
Majorana fermion chain experiences a spontaneous symmetry breaking and
localizes for repulsive interaction. To explain the instability of the critical
Majorana chain with respect to a combined effect of interaction and disorder,
we consider interaction as perturbation to the infinite-randomness fixed point
and calculate numerically two-wavefunction correlation functions that enter
interaction matrix elements. The numerical results supported by analytical
arguments exhibit a rich structure of critical eigenstate correlations. This
allows us to identify a relevant interaction operator that drives the Majorana
chain away from the infinite randomness fixed point. For the case of complex
fermions, the interaction is irrelevant.
