1205.1052 (Tieyan Si)

Braiding Majorana fermions on a triangular star model with frustrated
anyons    [PDF]

Tieyan Si

Few particle system with non-Abelian Majorana fermion is more likely to be implemented by quantum optical system. In order to show the non-Abelian particles and its corresponding spin configurations by conventional spin operators, we constructed a modified Kitaev honeycomb model of four particles with frustrated anyons on a triangular star. By analyzing the spin configurations of the eigenstates of this triangular star model, we find the product of a pair of Pauli matrices is one representation of anyon spin operators. The inverse Jordan-Wigner transformation had been applied to investigate the action of Majorana fermions upon the zero energy states. Composite Majorana fermion can stably exist at ground state while single Majorana fermion does not find pure ground state as its eigenstate. The Majorana fermions generate eigenstates out of vacuum. The eigenstates are the superposition states of zero energy states and the highest excited states. This is coincide with the non-Abelian fusion rules. We computed the braiding matrix for different ordering of the Majorana fermions which is encoded in the inverse Jordan-Wigner transformation. The non-commutable character of two neighboring braiding operations reveals the non-Abelian nature of these Majorana fermions.

View original: http://arxiv.org/abs/1205.1052