Interacting Weyl semimetals: characterization via the topological Hamiltonian and its breakdown

Weyl semimetals (WSMs) have robust linearly-dispersing excitations. Unusual properties arise from the latter, such as anomalous electrodynamic responses and open Fermi arcs on boundaries. We derive a simple criterion to identify and characterize WSMs in an interacting setting using the exact electronic Green's function at zero frequency, which defines a topological Bloch Hamiltonian. We apply this criterion by numerically analyzing, via cluster and other methods, interacting models with and without time-reversal symmetry. We thus identify mechanisms for how interactions move and renormalize Weyl fermions. Our methods remain valid in the presence of long-ranged Coulomb repulsion. Finally, we introduce a WSM-like phase for which our criterion breaks down, due to fractionalization of the electron.\\[4pt] W.~Witczak-Krempa, M.~Knap, D.~Abanin, Phys.~Rev.~Lett.~113, 136402 (2014)