Type-III and IV interacting Weyl points

3+1-dimensional Weyl fermions in interacting systems are described by effective quasi-relativistic Green's functions parametrized by a 16 element matrix $e^\mu_\alpha$ in an expansion around the Weyl point. The matrix $e^{\mu}_{\alpha}$ can be naturally identified as an effective tetrad field for the fermions. The correspondence between the tetrad field and an effective quasi-relativistic metric $g_{\mu\nu}$ governing the Weyl fermions allows for the possibility to simulate different classes of metric fields emerging in general relativity in interacting Weyl semimetals. According to this correspondence, there can be four types of Weyl fermions, depending on the signs of the components $g^{00}$ and $g_{00}$ of the effective metric. In addition to the conventional type-I fermions with a tilted Weyl cone and type-II fermions with an overtilted Weyl cone for $g^{00}>0$ and respectively $g_{00}>0$ or $g_{00}<0$, we find additional "type-III" and "type-IV" Weyl fermions with instabilities (complex frequencies) for $g^{00}<0$ and $g_{00}>0$ or $g_{00}<0$, respectively. While the type-I and type-II Weyl points allow us to simulate the black hole event horizon at an interface where $g^{00}$ changes sign, the type-III Weyl point leads to effective spacetimes with closed timelike curves.