  We investigate the Majorana fermions in a semiconductor nanostructure with
two wires connected through a ring. The nanostructure is mirror symmetric and
in the proximity of a superconductor. The Rashba spin-orbit coupling and a
magnetic field parallel to the wires or perpendicular to the ring are included.
Moreover, a magnetic flux is applied through the center of the ring, which
makes the phase difference of the superconducting order parameters in the two
wires being zero or $\pi$ due to the fluxoid quantization and the thermodynamic
equilibrium of the supercurrent in the superconducting ring. If the phase
difference is $\pi$, two Majorana modes are shown to appear around the ring
without interacting with each other. On contrast, if the phase difference is
zero, these Majorana modes disappear and the states localized around the ring
have finite energies. These states can be detected via the conductance
measurement by connecting two normal leads to the wires and a third one
directly to the ring. It is shown in the bias dependence of the differential
conductance from one of the leads connected to the wire to the one connected
directly to the ring that the tunnelings through the Majorana modes (i.e., in
the case with $\pi$ phase difference) leads to two peaks very close to the zero
bias, while the tunneling through the states with finite energies (i.e., in the
case with zero phase difference) leads to peaks far away from the zero bias if
the ring radius is small. This difference for the cases with and without the
Majorana modes in small ring radius is distinct and hence can be used to
identify the Majorana modes....
