  A large set of recent experiments has been exploring topological transport in
bosonic systems, e.g. of photons or phonons. In the vast majority,
time-reversal symmetry is preserved, and band structures are engineered by a
suitable choice of geometry, to produce topologically nontrivial bandgaps in
the vicinity of high-symmetry points. However, this leaves open the possibility
of large-quasimomentum backscattering, destroying the topological protection.
Up to now, it has been unclear what precisely are the conditions where this
effect can be sufficiently suppressed. In the present work, we introduce a
comprehensive semiclassical theory of tunneling transitions in momentum space,
describing backscattering for one of the most important system classes, based
on the valley Hall effect. We predict that even for a smooth domain wall
effective scattering centres develop at locations determined by both the local
slope of the wall and the energy. Moreover, our theory provides a quantitative
analysis of the exponential suppression of the overall reflection amplitude
with increasing domain wall smoothness.
