  We investigate a non-Hermitian Aharonov-Bohm (AB) ring system with a quantum
dot (QD) embedded in each of its two arms. The energy levels of the QDs are
complex in order to take into account the physical gain or loss of the ring
system during its interacting processes with the environment. When there is
magnetic flux threading through the ring, by allocating the flux phase factor
into the tunneling amplitudes between the QDs and the leads in different ways,
the Hamiltonian of the system can be written into different formalisms. We
calculate the transmission through the ring by using these different
non-Hermitian Hamiltonians and prove that it is not dependent on the way we
treat the phase factor, as in the Hermitian case. In addition, with appropriate
parameters, the asymmetric Fano profile will show up in the conductance
spectrum just by tuning the physical gain and loss of the system. The Fano
effect originates from the interferences of electrons traversing different
channels which are broadened or narrowed down due to the interaction between
the QDs and the environment. The proof we provide and the transport properties
revealed in this paper demonstrate the influences of the environment on an
otherwise isolated system and pave the way for the further studies on
non-Hermitian AB ring systems.
