The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal

The Adler-Bell-Jackiw (ABJ) axial anomaly is derived from the physical point of view as the production of Weyl particles and it is used to show the absence of the net production of particles for lattice regularized chirally invariant theories with locality. An analogy or a simulation is pointed out between the Weyl fermion theory and gapless semiconductors where two energy bands have pointlike degeneracies. For such materials, in the presence of parallel electric and strong magnetic fields, there exists an effect similar to the ABJ anomaly that is the movement of the electrons in the energy-momentum space from the neighborhood of one degeneracy point to another one. The longitudinal magneto-conduction becomes extremely strong.