# Type-III and IV interacting Weyl points

## TL;DR

This paper classifies interacting Weyl fermions into four types based on effective metric signatures, revealing new types (III and IV) with instabilities and implications for simulating gravitational phenomena in Weyl semimetals.

## Contribution

It introduces a novel classification of Weyl fermions into four types, including two new unstable types, based on effective metric signatures, expanding the understanding of Weyl semimetals and their gravitational analogs.

## Key findings

- Identification of four types of Weyl fermions based on effective metric components.
- Discovery of type-III and IV Weyl fermions with instabilities.
- Implications for simulating gravitational phenomena like black holes and closed timelike curves.

## Abstract

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.