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We investigate the strong-coupling phases that may arise in 3D Dirac and Weyl semimetals under the effect of the long-range Coulomb interaction, considering the many-body theory of these electron systems as a variant of the conventional fully relativistic Quantum Electrodynamics (QED). For this purpose, we apply two different nonperturba-tive approaches, consisting in the sum of ladder diagrams and taking the limit of a large number N of fermion flavors. We benefit from the renormalizability that the theory shows in both cases to compute the anomalous scaling dimensions of different operators exclusively in terms of the renormalized coupling constant, allowing us to determine the precise location of the singularities signaling the onset of the strong-coupling phases. We show then that the QED of 3D Dirac semimetals has two competing effects at strong coupling. One of them is the tendency to chiral symmetry breaking and dynamical mass generation, which are analogous to the same phenomena arising in the conventional QED at strong coupling. This trend is however outweighed by the strong suppression of electron quasiparticles that takes place at large N , leading to a different type of critical point at sufficiently large interaction strength, shared also by the 3D Weyl semimetals. Overall, the phase diagram of the 3D Dirac semimetals turns out to be richer than that of their 2D counterparts, displaying a transition to a phase with non-Fermi liquid behavior which may be observed in materials hosting a sufficiently large number of Dirac or Weyl points.

Strong-coupling phases of 3D Dirac and Weyl semimetals. A renormalization group approach