  We study the many-body physics in thin film topological band insulator, where
the inter-edge Coulomb interaction can lead to an exciton condensation
transition. We investigate the universality class of the exciton condensation
quantum critical point. With different chemical potentials and interactions,
the exciton condensation can belong to z = 2 mean field, or 3d XY, or
Yukawa-Higgs universality classes. The interplay between exciton condensate and
the time-reversal symmetry breaking is also discussed. Predictions of our work
can be tested experimentally by tuning the chemical potentials on both surfaces
of the thin film through gate voltage. We also show that all the analysis of
the exciton condensate can be directly applied to a spin-triplet superconductor
phase with attractive inter-edge interaction.
