The one-dimensional $p$-wave topological superconductor model with open-boundary conditions is examined in its topological phase. Using the eigenbasis of the noninteracting system I show that, provided the interactions are local and do not result in a closing of the gap, then even and odd parity sectors are unitarily equivalent. Following on from this, it is possible to define two many-body operators that connect each state in one sector with a degenerate counterpart in the sector with opposite parity. This result applies to all states in the system and therefore establishes, for a long enough wire, that all even-odd eigenpairs remain essentially degenerate in the presence of local interactions. Building on this observation I then set out a full definition of the related many-body Majorana operators and point out that their structure cannot be fully revealed using cross-correlation data obtained from the ground-state manifold alone. Although all results are formulated in the context of the one-dimensional $p$-wave model, I argue why they should also apply to more realistic realizations (e.g., the multichannel $p$-wave wire and proximity coupled models) of topological superconductivity.

• Received 31 October 2014
• Revised 30 March 2015

DOI:https://doi.org/10.1103/PhysRevB.92.081401