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We show that for closed finite sized systems with an odd number of realfermionic modes, even in the presence of interactions, there are always atleast two fermionic operators that commute with the Hamiltonian.There is a zeromode corresponding to the fermion parity operator, as shown by Akhmerov, aswell as additional linearly independent zero modes, one of which is 1) the onethat is continuously connected to the Majorana mode solution in thenon-interacting limit, and 2) less prone to decoherence when the system isopened to contact with an infinite bath. We also show that in the idealizedsituation where there are two or more well separated zero modes each associatedwith a finite number of fermions at a localized vortex, these modes havenon-Abelian Ising statistics under braiding. Furthermore the algebra of thezero mode operators makes them useful for fermionic quantum computation.

Exact Zero Modes in Closed Systems of Interacting Fermions