Xiong-Jun Liu, Chris L. M. Wong, K. T. Law
The study of non-Abelian Majorana zero modes advances our understanding of the fundamental physics in quantum matter, and pushes their potential applications to topological quantum computation. Recent investigations have shown that in the two-dimensional (2D) and 1D chiral superconductors, braiding isolated Majorana zero modes leads to noncommutative transformation of the quantum states \cite{Ivanov,Alicea1}. However, Majorana fermions in a $Z_2$ time-reversal invariant (TRI) topological superconductor come in pairs due to Kramers' theorem \cite{Ryu}. Therefore, braiding operations in TRI superconductors always exchange two pairs of Majorana fermions. In this work, we show surprisingly that, due to the protection of time-reversal symmetry, non-Abelian statistics can be obtained in the 1D TRI topological superconductor and may have advantages in applying to topological quantum computation. Furthermore, we predict a novel phenomenon in the Josephson effect for 1D TRI topological superconductors, which measures the topological qubit states in such systems.
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http://arxiv.org/abs/1304.3765
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