Non-conservation of the valley density and its implications for the observation of the valley Hall effect

We show that the conservation of the valley density in multi-valley and time-reversal-invariant insulators is broken in an unexpected way by the electric field that drives the valley Hall effect. This implies that fully-gapped insulators can support a valley Hall current in the bulk and yet show no valley density accumulation on the edges. Thus, the valley Hall effect cannot be observed in such systems. If the system is not fully gapped then valley density accumulation at the edges is possible and can result in a net generation of valley density. The accumulation has no contribution from undergap states and can be expressed as a Fermi surface average, for which we derive an explicit formula. We demonstrate the theory by calculating the valley density accumulations in an archetypical valley-Hall insulator: a gapped graphene nanoribbon. Surprisingly, we discover that a net valley density polarization is dynamically generated for some types of edge terminations.