Effects of broken time-reversal symmetry on transmission zeros

In this paper, we study the behavior of the transmission zeros and the corresponding change in the transmission phase when the time-reversal symmetry is broken by magnetic fields. For our study, we consider the closed Aharonov-Bohm(AB) interferometer with various embedded scattering centers. We find the following: (i) The transmission zeros are real when the AB flux is an integer or a half-integer multiple of the flux quantum, and the transmission phase jumps by $\pi$ at the zeros. (ii) The transmission zeros become complex or are shifted off the real-energy axis when the magnetic AB flux is not an integer or a half-integer multiple of the flux quantum, and the transmission phase evolves continuously. (iii) The distance of the zeros from the real-energy axis or the imaginary part of the transmission zeros is sinusoidal as a function of the magnetic AB phase. We suggest the experimental setup which can test our results.