Signature of indirect magnetic interaction in the crossover from type-I to type-II Weyl semimetals

We have theoretically investigated the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in Weyl semimetals (WSMs) with tilted dispersion in the framework of the Matsubara Green's function. It is shown that there emerges a sharp peak or dip of the RKKY interaction at the phase boundary, which can unambiguously characterize the phase transition from type-I to type-II WSMs. Distinct responses of the RKKY interaction to the tilted parameter $A$ in two phases are revealed to distinguish different Weyl phases. Moreover, we find that a nonzero Dzyaloshinskii-Moriya term survives even for Fermi energy at the Weyl points, which is completely contributed by the tilting effect and is distinguished from the previous mechanism. Consequently, the RKKY interaction provides richer signatures superior to usual conductivity to probe Weyl phases with tilted dispersion since it involves a contribution from both the Fermi surface and the Fermi sea. We have performed an analysis with both the analytical formula and numerical calculations.