  We study one-component fermions in chain lattices with proximity-induced
superconducting gap and interparticle short-range interaction, capable of
hosting Majorana fermions. By systematically tracking various physical
quantities, we show that topological states and topological phase transitions
in the system can be identified by multiple signatures in thermodynamic
quantities and pair-condensate properties, in good agreement with the known
signatures in the ground-state energy and entanglement spectrum. We find the
disappearance of the topological phase in a largely attractive regime, in which
the system undergoes a first-order transition between two topologically trivial
states. In addition, the stability of the signatures against finite size,
disorder, and inhomogeneity is analyzed. Our results provide additional degrees
of freedom for the characterization of topological states with interaction and
for the experimental detection of emergent Majorana fermions.
