High-dimensional disorder-driven phenomena in nodal semimetals and systems with long-range hopping

Systems of non-interacting electrons are believed widely to exhibit only one type of disorder-driven transitions: the Anderson localisation transition. It has been suggested, however, that systems with the power-law quasiparticle dispersion k^α in high dimensions d>2α, exemplified by 3D Weyl and Dirac semimetals, may exhibit transitions in a different universality class, as well as unconventional energy-level statistics, Lifshitz tails and ballistic-transport properties. In this talk, I will review existing results on the non-Anderson transitions and other unconventional disorder-driven phenomena in nodal semimetals and related systems (quantum kicked rotors, arrays of ultracold ions, 1D and 2D plasmonic systems, etc.). Also, I will demonstrate that the field theories of disordered nodal semimetals with α<d can be mapped exactly onto those of systems with long-range hopping of quasiparticles, where hopping decays with distance r slower than 1/r^d (trapped ultracold ions, spins in solids, nitrogen defects in diamonds, etc.). This duality allows to describe the properties of each of these two classes of systems using the results established for the other class and, in particular, establishes the existence of unconventional disorder-driven transitions in systems with long-range hopping.