Abstract

We discover that the conservation of the valley density in multi-valley insulators is broken in an unexpected way by the electric field that drives the valley Hall effect in the first place. This implies that time-reversal-invariant fully-gapped insulators, in which no bulk or edge state crosses the Fermi level, can support a valley Hall current in the bulk and yet show no valley density accumulation at the edges. If the system is not fully gapped then valley density accumulation at the edges is possible. The accumulation has no contribution from undergap states and can be expressed as a Fermi surface average. We also discover that a net valley density polarization is dynamically generated for certain edge terminations. We draw general conclusions which apply to other phenomena, such as the orbital Hall effect.