  The interplay of band topology and electron interactions can lead to novel
quantum states of matter. In this work, we theoretically study the quantum spin
Hall insulator (QSHI) in a perpendicular magnetic field. In the noninteracting
case, the QSHI with space inversion and uniaxial spin rotation symmetries
undergoes a topological transition into a normal insulator phase at a critical
magnetic field $B_{\mathrm{c}}$. The exciton condensation in the lowest Landau
levels is triggered by Coulomb interactions in the vicinity of $B_{\mathrm{c}}$
and spontaneously breaks the inversion and the spin rotation symmetries. We
propose that the electron spin resonance spectroscopy can directly probe the
exciton condensation order. Our results should apply to the InAs/GaSb quantum
wells and other QSHIs.
