Many-body Majorana operators and the equivalence of parity sectors

TL;DR

This paper demonstrates that in a 1D p-wave topological superconductor, local interactions do not break the degeneracy between even and odd parity sectors, allowing the definition of many-body Majorana operators that connect degenerate states.

Contribution

It introduces a framework for defining many-body Majorana operators in interacting systems and proves the parity sector equivalence under local interactions without gap closing.

Findings

Even and odd parity sectors are unitarily equivalent with local interactions.

Degeneracy between eigenstates persists in the presence of local interactions.

Many-body Majorana operators can be explicitly constructed in these systems.

Abstract

The one-dimensional p-wave topological superconductor model with open-boundary conditions is examined in its topological phase. Using the eigenbasis of the non-interacting 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 1-dimensional p-wave model, I argue why they should also apply to more realistic realisations (e.g. the multi-channel p-wave wire and proximity coupled models) of topological superconductivity.