Axial anomaly in multi-Weyl and triple-point semimetals

We derive the expression of the abelian axial anomaly in the so-called multi- Weyl and triple-point crossing semimetals. No simplifying restrictions are assumed on the symmetry of the spectrum. Three different computation methods are considered: the per- turbative quantum field theory procedure which is based on the evaluation of the one-loop Feynman diagrams, the Nielsen-Ninomiya method, and the Atiyah-Singer index argument. It is shown that the functional form of the axial anomaly does not depend on the Lorentz symmetry, but it is determined by the gauge structure group. We discuss the stability of the anomaly — stemming from the quantisation of the anomaly coefficient — under smooth modifications of the lagrangian parameters.