Quantum Lifshitz criticality of topological semimetals

In the absence of particle-hole symmetry, all clean topological semimetals can be realized in two thermodynamically distinct forms: a power-law, incompressible phase and a compressible phase, respectively possessing zero and constant density of states at the band touching energy. Therefore, the density of states can serve as an order parameter for describing quantum Lifshitz transitions between two thermodynamically distinct but topologically equivalent states of matter. We address the nature of such quantum Lifshitz transitions for different topological semimetals, and show how they can strongly influence thermodynamic, spectroscopic, and transport properties of many quantum materials.