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Vortex Loops and Majoranas

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.