Exact Zero Modes in Closed Systems of Interacting Fermions

TL;DR

This paper demonstrates the existence of multiple zero modes in finite closed systems of interacting fermions, including their algebraic properties and potential for topological quantum computation.

Contribution

It establishes the presence of multiple zero modes in interacting fermionic systems with odd modes, extending known results to interacting cases and exploring their non-Abelian statistics.

Findings

Existence of at least two zero modes in finite systems with odd fermion number.

Zero modes include the fermion parity operator and additional independent modes.

Zero modes exhibit non-Abelian Ising statistics under braiding.

Abstract

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