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Physics Letters A

New implementation of composite fermions in terms of subgroups of a braid group

Janusz Edward Jacak, Ireneusz Jóźwiak, Lucjan Jacak

Published Nov 4, 2009• 15 citations
Publisher
DOI
10.1016/j.physleta.2009.10.075

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A new implementation of composite fermions, and more generally — of composite anyons is formulated, exploiting one-dimensional unitary representations of appropriately constructed subgroups of the full braid group, in accordance with a cyclotron motion of 2D charged particle systems. The nature of hypothetical fluxes attached to the Jain's composite fermions is explained via additional cyclotron trajectory loops consistently with braid subgroup structure. It is demonstrated that composite fermions and composite anyons are rightful 2D particles (not an auxiliary construction) associated with cyclotron braid subgroups instead of the full braid group, which may open a new opportunity for non-Abelian composite anyons for quantum information processing applications.
It is shown that composite fermions—particles which appear in fractional quantum Hall effect, and are statistically different from anyons—could be described within braid group formalism. In the proposed model some ways of exchange of bare particles of composite particles are restricted. The new group describing exchanges of particles is defined. The description of such a group is included and the unitary one-dimensional representation of this group is found. An explicit form of this group for two exemplary cases is presented.
Three types of traditional hard problem could be resolved by Shor,Boneh and Lipton's quantum algorithms in po-lynomial time.By the braid group constructed a lot of mathematics difficulties were not an effective solution under the conditions of the quantum computer.It seemed that braid group was a kind of considerable considerable cryptography platform in the future.This paper proposed a new proxy blind signature scheme based on conjugate search problem and the p-th root finding problem,and the exchangeable of the group operation between the elements in the left subgroup and the right subgroup of a braid group.Through program analysis shows that the new scheme is effective and feasible.