Topological Invariants of Metals and 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 non-zero for each of its Fermi surfaces. We show that such an invariant can be related to 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 the effect is dictated by a second class Chern invariant. These connections open the pathway to observe the hidden topological invariants in metallic systems. 
Jianhui Zhou, Hua Jiang, Qian Niu, Junren Shi
cond-mat.mes-hall cond-mat.str-el