Topological Invariants of Metals and the Related Physical Effects

The total reciprocal space magnetic flux threading through a closed Fermi surface is a topological invariant for a three-dimensional metal. For a Weyl metal, the invariant is nonzero for each of its Fermi surfaces. We show that such an invariant can be related to the magneto-valley-transport effect, in which an external magnetic field can induce a valley current. We further show that a strain field can drive an electric current, and that the effect is dictated by a second-class Chern invariant. These connections open the pathway to observe the hidden topological invariants in metallic systems.