Interplay of quantum spin Hall effect and spontaneous time-reversal symmetry breaking in electron-hole bilayers. II. Zero-field topological superconductivity

Funding Information: We thank D. I. Pikulin for useful discussions and comments. The paper is supported by the Foundation for Polish Science through the IRA Programme co-financed by EU within SG OP and the Academy of Finland Project No. 331094. We acknowledge the computational resources provided by the Aalto Science-IT project and access to the computing facilities of the Interdisciplinary Center of Modeling at the University of Warsaw, Grants No. GB82-13, No. G78-13 and No. G75-10. Publisher Copyright: © 2022 American Physical Society.It has been proposed that band-inverted electron-hole bilayers support a phase transition from an insulating phase with spontaneously broken time-reversal symmetry to a quantum spin Hall insulator phase as a function of increasing electron and hole densities. Here we show that in the presence of proximity-induced superconductivity, it is possible to realize Majorana zero modes in the time-reversal symmetry broken phase in the absence of magnetic field. We develop an effective low-energy theory for the system in the presence of a time-reversal symmetry-breaking order parameter to obtain analytically the Majorana zero modes and we find good agreement between the numerical and analytical results in the limit of weakly broken time-reversal symmetry. We show that the Majorana zero modes can be detected in superconductor/time-reversal symmetry broken insulator/superconductor Josephson junctions through the measurement of a 4π Josephson current. Finally, we demonstrate that the Majorana fusion-rule detection is feasible by utilizing the gate voltage dependence of the spontaneous time-reversal symmetry breaking order parameter.Peer reviewe