Controlling Fusion of Majorana Fermions in one-dimensional systems by Zeeman Field

We propose to realize Majorana fermions (MFs) on an edge 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 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.