Seiberg-Witten monopoles: Weyl metal coupled to chiral magnets

We study a Weyl (semi)metal which couples to local magnets. In the continuum limit, the Hamiltonian of the system matches the Chern-Simons-Maxwell-Dirac functional and then the ground state is governed by generalized Seiberg-Witten (SW) or Freund equations in terms of the sign of Dzyaloshinskii-Moriya coupling. The ground states determined by the Freund equations may either be monopolar Weyl semimetal accompanied by the ferromagnetic magnets (MWFM) or SW monopoles which consist of spheric Weyl fermions coupled to chiral magnets, depending on the strength of the Kondo coupling. In the latter phase, the topological ground state is characterized by SW invariants and with a Weyl surface on which the Weyl metal is of an exotic dispersion $\propto \sqrt k$. There are also the metastable SW monopole solutions carrying an opposite SW invariant for the SW equations while the ground state in this case is the MWFM state.