Controlling Fusion of Majorana Fermions in One-Dimensional Systems by Zeeman Field

We propose the realization of Majorana fermions (MFs) on the edges of a two-dimensional topological insulator in the proximity with s-wave superconductors and in the presence of transverse exchange field h. It is shown that there appear a pair of MFs localized at two junctions and that a reverse in the direction of h can lead to permutation of two MFs. With decreasing h, the MF states can either be fused or form one Dirac fermion on the \pi-junctions, exhibiting a topological phase transition. This characteristic can be used to detect physical states of MFs when they are transformed into Dirac fermions localized on the \pi-junction. A condition of decoupling two MFs is also given.