Exotic Lifshitz transitions in topological materials

Topological Lifshitz transitions involve many types of topological structures in momentum and frequency–momentum spaces, such as Fermi surfaces, Dirac lines, Dirac and Weyl points, etc., each of which has its own stability-supporting topological invariant ($N_1$, $N_2$, $N_3$, ${\tilde N}_3$, etc.). The topology of the shape of Fermi surfaces and Dirac lines and the interconnection of objects of different dimensionalities produce a variety of Lifshitz transition classes. Lifshitz transitions have important implications for many areas of physics. To give examples, transition-related singularities can increase the superconducting transition temperature; Lifshitz transitions are the possible origin of the small masses of elementary particles in our Universe, and a black hole horizon serves as the surface of the Lifshitz transition between vacua with type-I and type-II Weyl points.