  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 time-reversal symmetry breaking order parameter to obtain analytically the
Majorana zero modes and we find a 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 $4\pi$ 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.
