Type-I and type-II Weyl fermions, topological depletion, and universal subleading scaling across topological phase transitions
AbstractIt is well established that physical quantities satisfy scaling functions across a quantum phase transition with an order parameter. It remains an open problem if there are scaling functions across a topological quantum phase transition (TQPT) with extended Fermi surfaces (FSs). Here, we study a simple system of fermions hopping in a cubic lattice subject to Weyl-type spin-orbit coupling (SOC). As one tunes the SOC parameter at half filling, the system displays both type-I and type-II Weyl fermions and also various TQPTs driven by the collision of particle-particle or hole-hole Weyl FSs. At zero temperature, the TQPT is found to be third order, and its critical exponents are determined. Then we investigate if the physical quantities such as specific heat, compressibility, and magnetic susceptibilities satisfy any sort of scaling across the TQPT. In contrast to all the previous cases in quantum or topological transitions, we find that although the leading terms are nonuniversal and cutoff dependent, the subleading terms are nonanalytic and satisfy universal scaling relations. The subleading scaling leads to topological depletions which show non-Fermi-liquid corrections and T quantum cusps. One can also form a topological Wilson ratio from the subleading scalings of two conserved quantities such as the specific heat and the compressibility. One may also interpret the type-I and type-II Weyl fermions as a TQPT driven by the collision of particle-hole Weyl FSs. Experimental realizations and detections in cold-atom systems and materials with SOC are discussed.