General formula of chiral anomaly for type-I and type-II Weyl semimetals

Weyl semimetals (WSMs) are classified into type-I and type-II, depending on the magnitudes of the inclination of Weyl cones. It is known that these WSMs show negative longitudinal magnetoresistance originating from chiral anomaly. Moreover, we have recently revealed that type-II WSMs show positive longitudinal magnetoresistance originating from chiral anomaly. The negative longitudinal magnetoresistance in type-I WSMs can be explained utilizing the conventional formula of the chiral anomaly, which does not have the term related to the inclination of the Weyl cones. However, we cannot explain both the positive and the negative longitudinal magnetoresistance in type-II WSMs utilizing it. Therefore, in this paper, we derive the general formula including the term related to the inclination of the Weyl cones in order to explain straightforwardly the positive and the negative longitudinal magnetoresistance in type-II WSMs. Also, we consider both cases where a pair of the Weyl cones are tilted in the same direction (positive tilt chirality) and toward (or against) each other (negative tilt chirality) in order to investigate the influence of the direction to which the Weyl cones are tilted. As a result, we find that in the negative tilt chirality, the general formula is strongly affected by the inclination. These results suggest that we can estimate whether the WSMs show the positive or the negative longitudinal magnetoresistance using the general formula from the information of their tilt chirality and the magnitudes of the inclination of the Weyl cones.