Realization of the Non-endpoint Majorana Bound States in an Extended Kitaev Chain

We study the energy levels and transport properties of an extended Kitaev chain with a phase gradient. It is demonstrated that the hopping phase difference can effectively induce the generation of Majorana bound states, which are located at the non-endpoint sites of the chain. The number and positions of the non-endpoint Majorana bound states can be modulated by the hopping phase difference and initial hopping phase, respectively. In addition, we propose a protocol to realize topological braiding operation by exchanging the positions of two Majorana bound states in the extended Kitaev ring. Furthermore, we also implement the braiding of any two of the multiple Majorana bound states in the extended Kitaev double rings.