Modeling electron fractionalization with unconventional Fock spaces

TL;DR

This paper introduces a novel approach to modeling fractionally-charged quasiparticles using unconventional Fock algebras derived from roots of fermionic operators, connecting to parafermion modes and fractional quantum Hall states.

Contribution

It develops a framework for fractionalizing fermionic charges via root-based Fock algebras, linking mathematical structures to physical quasiparticles and zero-energy modes.

Findings

Fermion-root quasiparticles carry fractional charge and spin.

Multiple exchange statistics arise from non-uniqueness of fermion roots.

Numerical analysis shows hybridization of Majorana and parafermion modes.

Abstract

It is shown that certain fractionally-charged quasiparticles can be modeled on \(D-\)dimensional lattices in terms of unconventional yet simple Fock algebras of creation and annihilation operators. These unconventional Fock algebras are derived from the usual fermionic algebra by taking roots (the square root, cubic root, etc.) of the usual fermionic creation and annihilation operators. If the fermions carry non-Abelian charges, then this approach fractionalizes the Abelian charges only. In particular, the \(m\)th-root of a spinful fermion carries charge \(e/m\) and spin \(1/2\). Just like taking a root of a complex number, taking a root of a fermion yields a mildly non-unique result. As a consequence, there are several possible choices of quantum exchange statistics for fermion-root quasiparticles. These choices are tied to the dimensionality \(D=1,2,3,\dots\) of the lattice by basic physical considerations. One particular family of fermion-root quasiparticles is directly connected to the parafermion zero-energy modes expected to emerge in certain mesoscopic devices involving fractional quantum Hall states. Hence, as an application of potential mesoscopic interest, I investigate numerically the hybridization of Majorana and parafermion zero-energy edge modes caused by fractionalizing but charge-conserving tunneling.