  The topological states of matter and topological materials have been
attracting extensive interests as one of the frontier topics in condensed
matter physics and materials science since the discovery of quantum Hall effect
in 1980s. So far all the topological phases such as quantum Hall effect,
quantum spin Hall effect and topological insulators and superconductors are
characterized by a nonzero integer or Z and Z2 topological invariant. None is a
half-integer or fractional. Here we propose a novel type of semimetals which
hosts a single cone of Wilson fermions instead of Dirac fermions. The Wilson
fermions possess linear dispersion near the energy crossing point, but breaks
the chiral or parity symmetry such that an unpaired Dirac cone can be realized
on a lattice. They are not prohibited by the Nielsen-Ninomiya theorem and avoid
the fermion doubling problem. We find that the system can be classified by the
relative homotopy group, and the topological invariant is a half-integer. We
term the unexpected and nontrivial quantum phase as "quantum anomalous
semimetal". The topological phase is a synergy of topology of band structure in
solid and quantum anomaly in quantum field theory. The work opens the door
towards exploring novel states of matter with fractional topological charge.
