  We investigate the role that vortex loops play in characterizing eigenstates
of interacting Majoranas. We first give some general results, and then we focus
on ladder Hamiltonian examples to test further ideas. Two methods yield exact
results: i.) We utilize the mapping of spin Hamiltonians to quartic
interactions of Majoranas and show under certain conditions the spectra of
these two examples coincide. ii) In cases with reflection-symmetric
Hamiltonians, we use reflection positivity for Majoranas to characterize
vortices. Aside from these exact results, two additional methods suggest wider
applicability of these results: iii.) Numerical evidence suggests similar
behavior for certain systems without reflection symmetry. iv.) A perturbative
analysis also suggests similar behavior without the assumption of reflection
symmetry.
