  The interplay of electronic band structures and electron-electron
interactions is known to brew new phases in condensed matter. In this paper, we
investigate thermodynamic phases and corresponding electronic structures of the
Weyl semimetal in the strong onsite Coulomb interaction limit. Based on a
minimum model of the Weyl semimetal with two linear Weyl nodes, it is shown
that generically the Weyl semimetal becomes magnetic in the presence of
interactions. In particular, it is shown that the Dzyaloshinskii-Moriya
exchange interaction is generally induced so that the A-type antiferromagnetic
(A-AFM) phase and the spiral spin density wave (SSDW) states are two generic
phases. Furthermore, we find that Weyl nodes proliferate and it is possible to
doubly enhance the unusual properties of non-interacting Weyl semimetals
through the realization of double-Weyl nodes in strong correlation limit.
Specifically, it is shown that in the SSDW phase, linear Weyl nodes are tuned
into double-Weyl nodes with the corresponding charges being $\pm 2$. As the
spin-orbit coupling increases, a quantum phase transition occurs with the SSDW
phase being turned into an A-AFM phase and at the same time, double-Weyl nodes
are disintegrated into two pairs of linear Weyl nodes. Our results reveal the
unusual interplay between the topology of electronic structures and magnetism
in strongly correlated phases of Weyl semimetals.
