Topological density-wave states in a particle-hole symmetric Weyl metal

We study the instabilities of a particle-hole symmetric Weyl metal with both electron and hole Fermi surfaces (FS) around the Weyl points. For a repulsive interaction, we find that the leading instability is towards a longitudinal spin-density-wave order (SDW$_z$). Besides, there exist three degenerate subleading instabilities: a charge-density-wave (CDW) instability and two transverse spin-density-wave (SDW$_{x,y}$) instabilities. For an attractive interaction the leading instabilities are towards two pair-density-wave orders (PDW) which pair the two FS's separately. Both the PDW and SDW$_z$ order parameters fully gap out FS's, while the CDW and SDW$_{x,y}$ ones leave line nodes on both FS's. For the SDW$_z$ and the PDW states, the surface Fermi arc in the metallic state evolves to a chiral Fermi line which passes the projection of the Weyl points and traverses the full momentum space. For the CDW state, the line node projects to a "drumhead" band localized on the surface, which can lead to a topological charge polarization. We verify the surface states by numerically simulating the angular-resolved photoemission spectroscopy data.