# Exotic Lifshitz transitions in topological materials

## TL;DR

This paper explores various topological Lifshitz transitions in materials, highlighting their complex structures, invariants, and implications across physics, including superconductivity, particle physics, and black hole horizons.

## Contribution

It provides a comprehensive analysis of topological Lifshitz transitions, classifying different types and discussing their physical consequences across multiple fields.

## Key findings

- Multiple classes of Lifshitz transitions identified.
- Topological invariants support stability of structures.
- Transitions influence phenomena like superconductivity and particle masses.

## Abstract

Topological Lifshitz transitions involve many types of topological structures in momentum and frequency-momentum spaces: Fermi surfaces, Dirac lines, Dirac and Weyl points, etc. Each of these structures has their own topological invariant ($N_1$, $N_2$, $N_3$, $\tilde N_3$, etc.), which supports the stability of a given topological structure. The topology of the shape of Fermi surfaces and Dirac lines, as well as the interconnection of the objects of different dimensions, lead to numerous classes of Lifshitz transitions. The consequences of Lifshitz transitions are important in different areas of physics. The singularities emerging at the transition may enhance the transition temperature to superconductivity; the Lifshitz transition can be in the origin of the small masses of elementary particles in our Universe; the black hole horizon serves as the surface of Lifshitz transition between the vacua with type-I and type-II Weyl points; etc.