# Integer quantum Hall conductance and time-reversal symmetry

If we have a (2+1)-dimensional electronic gapped system with a unique ground state and it has a nonzero integer quantum Hall conductance, then the system (or its ground state) must break the time-reversal symmetry explicitly.

My question is how about its converse and to what extent the converse holds. Since the Chern-Simons is the only relevant characterizing response theory at an infinite-length scale in (2+1) dimensions, my guess is that a vanishing quantum Hall conductance sufficiently implies that the unique ground state is time-reversal symmetric at the low-energy (or long-range) limit. Is my guess correct or how to prove it? Can we further generalize to general ground-state degeneracies?

There are other long-wavelength responses that break time-reversal symmetry other than Hall conductance, for example thermal Hall conductance. I believe Hall conductance and thermal Hall conductance are the only ones for a system with only $$U(1)$$ symmetry that has no fractionalized excitations (e.g. anyons), but if you have anyonic excitations, then their braiding statistics can also violate time-reversal symmetry, for example.