  Fermi liquid theory works very well in most normal metals, but is found
violated in many strongly correlated electron systems, such as cuprate and
heavy-fermion superconductors. A widely accepted criterion is that, the Fermi
liquid theory is valid when the interaction-induced fermion damping rate
approaches zero more rapidly than the energy. Otherwise, it is invalid. Here,
we demonstrate that this criterion breaks down in topological double-and
triple-Weyl semimetals. Renormalization group analysis reveals that, although
the damping rate of double- and triple-Weyl fermions induced by the Coulomb
interaction approaches zero more rapidly than the energy, the quasiparticle
residue vanishes and the Fermi liquid theory is invalid. This behavior
indicates a weaker-than-marginal violation of the Fermi liquid theory. Such an
unconventional non-Fermi liquid state originates from the special dispersion of
double- and triple-Weyl fermions, and is qualitatively different from all the
other Fermi-liquid and non-Fermi-liquid states. The predicted properties of the
fermion damping rate and the spectral function can be probed by the
angle-resolved photoemission spectroscopy. The density of states, specific
heat, and conductivities are also calculated and analyzed after incorporating
the corrections induced by the Coulomb interaction.
