  Three-dimensional Weyl and Dirac semimetals can support a
chiral-symmetry-breaking, fully gapped, charge-density-wave order even for
sufficiently weak repulsive electron-electron interactions, when placed in
strong magnetic fields. In the former systems, due to the natural momentum
space separation of Weyl nodes the ordered phase lacks the translational
symmetry and represents an axionic phase of matter, while that in a Dirac
semimetal (neglecting the Zeeman coupling) is only a trivial insulator. We
present the scaling of this spectral gap for a wide range of subcritical (weak)
interactions as well as that of the diamagnetic susceptibility with the
magnetic field. A similar mechanism for charge-density-wave ordering at weak
coupling is shown to be operative in double and triple-Weyl semimetals, where
the dispersion is linear (quadratic and cubic, respectively) for the z (planar)
component(s) of the momentum. We here also address the competition between the
charge-density-wave and a spin-density-wave orders, both of which breaks the
chiral symmetry and leads to gapped spectrum, and show that at least in the
weak coupling regime the former is energetically favored. The anomalous surface
Hall conductivity, role of topological defects such as axion strings, existence
of one-dimensional gapless dispersive modes along the core of such defects, and
anomaly cancellation through the Callan-Harvey mechanism are discussed.
